Search: a123484 -id:a123484
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A033762
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Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).
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+0
39
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1, 1, 0, 2, 1, 0, 2, 0, 0, 2, 2, 0, 1, 1, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 4, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 2, 0
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OFFSET
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0,4
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COMMENTS
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Number of solutions of 8*n + 4 = x^2 + 3*y^2 in positive odd integers. - Michael Somos, Sep 18 2004
Half the number of integer solutions of 4*n + 2 = x^2 + y^2 + z^2 where 0 = x + y + z and x and y are odd. - Michael Somos, Jul 03 2011
Given g.f. A(x), then q^(1/2) * 2 * A(q) is denoted phi_1(z) where q = exp(Pi i z) in Conway and Sloane.
Half of theta series of planar hexagonal lattice (A2) with respect to an edge.
Bisection of A002324. Number of ways of writing n as a sum of a triangular plus three times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011
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REFERENCES
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Burce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 223 Entry 3(i).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 103. See Eq. (13).
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.27).
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LINKS
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FORMULA
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Expansion of q^(-1/2) * (eta(q^2) * eta(q^6))^2 / (eta(q) * eta(q^3)) in powers of q. - Michael Somos, Apr 18 2004
Expansion of q^(-1) * (a(q) - a(q^4)) / 6 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Oct 24 2006
Expansion of psi(x) * psi(x^3) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 03 2011
Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, -2, ...]. - Michael Somos, Apr 18 2004
Given g.f. A(x), then B(x) = (x * A(x^2))^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 4*u*v*w + 16*v*w^2 - 8*w*v^2 - w*u^2.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p==5 (mod 6) otherwise b(p^e) = e+1. (Clarification: the g.f. A(x) is not the primary function of interest, but rather B(x) = x * A(x^2), which is an eta-quotient and is the generating function of a multiplicative sequence.)
G.f.: (Sum_{j>0} x^((j^2 - j) / 2)) * (Sum_{k>0} x^(3(k^2 - k) / 2)) = Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k * (1 - x^k) * (1 - x^(4*k)) * (1 - x^(5*k)) / (1 - x^(12*k)). (End)
G.f.: s(4)^2*s(12)^2/(s(2)*s(6)), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - x^(4*k) / (1 + x^(4*k) + x^(8*k)). - Michael Somos, Nov 04 2005
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 23 2023
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EXAMPLE
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G.f. = 1 + x + 2*x^3 + x^4 + 2*x^6 + 2*x^9 + 2*x^10 + x^12 + x^13 + 2*x^15 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 2*q^13 + 2*q^19 + 2*q^21 + q^25 + q^27 + 2*q^31 + ...
a(6) = 2 since 8*6 + 4 = 52 = 5^2 + 3*3^2 = 7^2 + 3*1^2.
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Jul 03 2011 *)
QP = QPochhammer; s = (QP[q^2]*QP[q^6])^2/(QP[q]*QP[q^3]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(2 n + 1))]; (* Michael Somos, Mar 06 2016 *)
%t A033762 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}]; (* Michael Somos, Mar 06 2016 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 18 2004 */
(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -12, d) * (n / d % 2)))}; /* Michael Somos, Nov 04 2005 */
(PARI) {a(n) = if( n<0, 0, n = 8*n + 4; sum( j=1, sqrtint( n\3), (j%2) * issquare(n - 3*j^2)))} /* Michael Somos, Nov 04 2005 */
(PARI) {a(n) = if( n<0, 0, sumdiv(2*n + 1, d, kronecker(-3, d)))}; /* Michael Somos, Mar 06 2016 */
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 202); A[2] + A[4]; /* Michael Somos, Jul 25 2014 */
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CROSSREFS
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Cf. A002324, A004016, A005881, A035178, A091393, A093766, A093829, A096936, A112298, A113447, A113661, A113974, A115979, A122860, A123331, A123484, A136748, A137608.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A112607
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Number of representations of n as a sum of a triangular number and twelve times a triangular number.
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15
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1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0
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OFFSET
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0,16
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COMMENTS
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Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
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FORMULA
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a(n) = 1/2*( d_{1, 3}(8n+13) - d_{2, 3}(8n+13) ) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-13/8)*(eta(q^2)*eta(q^24))^2/(eta(q)*eta(q^12)) in powers of q. - Michael Somos, Sep 29 2006
Expansion of psi(q)*psi(q^12) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Sep 29 2006
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos, Sep 29 2006
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EXAMPLE
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a(15) = 2 since we can write 15 = 15 + 12*0 = 3 + 12*1.
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MATHEMATICA
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a[n_] := DivisorSum[8n+13, KroneckerSymbol[-3, #]&]/2; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)
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PROG
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(PARI) {a(n)=if(n<0, 0, n=8*n+13; sumdiv(n, d, kronecker(-3, d))/2)} /* Michael Somos, Sep 29 2006 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^24+A)^2/eta(x+A)/eta(x^12+A), n))} /* Michael Somos, Sep 29 2006 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A112609
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Number of representations of n as a sum of three times a triangular number and four times a triangular number.
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14
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1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0
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OFFSET
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0,31
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COMMENTS
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Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
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M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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LINKS
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FORMULA
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a(n) = 1/2*( d_{1, 3}(8n+7) - d_{2, 3}(8n+7) ) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of phi(q^3) * psi(q^4) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Mar 10 2008
Expansion of q^(-7/8) * (eta(q^6) * eta(q^8))^2 / (eta(q^3) * eta(q^4)) in powers of q. - Michael Somos, Mar 10 2008
Euler transform of period 24 sequence [ 0, 0, 1, 1, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 0, -2, ...]. - Michael Somos, Mar 10 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138270.
a(3*n+2) = 0.
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EXAMPLE
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a(30) = 2 since we can write 30 = 3*10 + 4*0 = 3*6 + 4*3
q^7 + q^31 + q^39 + q^63 + q^79 + q^103 + q^111 + q^127 + q^151 + ...
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MATHEMATICA
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A112609[n_] := SeriesCoefficient[(QPochhammer[q^6]*QPochhammer[q^8])^2/
(QPochhammer[q^3]*QPochhammer[q^4]), {q, 0, n}]; Table[A112609[n], {n, 0, 50}] (* G. C. Greubel, Sep 25 2017 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n=8*n+7; sumdiv(n, d, kronecker(-3, d))/2)} /* Michael Somos, Mar 10 2008 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^8 + A))^2 / (eta(x^3 + A) * eta(x^4 + A)), n))} /* Michael Somos, Mar 10 2008 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A112606
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Number of representations of n as a sum of six times a square and a triangular number.
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13
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1, 1, 0, 1, 0, 0, 3, 2, 0, 2, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 3, 0, 0, 2, 2, 0, 4, 1, 0, 2, 0, 0, 0, 4, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 0, 2, 0, 0, 4, 2, 0, 0, 2, 0, 1, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 2, 4, 0, 4, 0, 0, 4, 0, 0
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OFFSET
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0,7
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COMMENTS
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The greedy inverse starts 2, 0, 7, 6, 27, 300, 349, 14706, 216, 1035, 17107,... - R. J. Mathar, Apr 28 2020
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REFERENCES
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M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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LINKS
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FORMULA
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a(n) = d_{1, 3}(8n+1) - d_{2, 3}(8n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^12)^5 /(eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q. - Michael Somos, Sep 29 2006
Expansion of phi(q^6) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -4, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos, Sep 29 2006
G.f.: (Sum_{k} x^(6*k^2)) * (Sum_{k>0} x^((k^2-k)/2)). a(3*n+2)=0. - Michael Somos, Sep 29 2006
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EXAMPLE
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1 + x + x^3 + 3*x^6 + 2*x^7 + 2*x^9 + x^10 + 2*x^12 + x^15 + 2*x^16 + ...
q + q^9 + q^25 + 3*q^49 + 2*q^57 + 2*q^73 + q^81 + 2*q^97 + q^121 + 2*q^129 + ...
a(6) = 3 since we can write 6 = 6*1^2 + 0 = 6*(-1)^2 + 0 = 0 + 6.
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ 8 n + 1]}]] (* Michael Somos, Jun 16 2011 since V6 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 3, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]] (* Michael Somos, Jun 16 2011 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; sumdiv(n, d, kronecker(-3, d)))} /* Michael Somos, Sep 29 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))} /* Michael Somos, Sep 29 2006 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A131961
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Expansion of f(x, x^2) * f(x^2, x^2) in powers of x where f(, ) is Ramanujan's general theta function.
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+0
12
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1, 1, 3, 2, 2, 1, 0, 3, 2, 4, 2, 0, 1, 2, 2, 3, 0, 2, 2, 2, 4, 0, 1, 4, 2, 2, 1, 0, 2, 0, 4, 0, 2, 4, 4, 1, 0, 4, 0, 2, 3, 0, 2, 2, 4, 0, 0, 2, 2, 0, 2, 3, 2, 4, 2, 2, 0, 3, 4, 4, 0, 0, 2, 0, 0, 4, 0, 2, 0, 2, 1, 0, 8, 2, 2, 2, 2, 3, 2, 4, 0, 0, 0, 2, 2, 4, 0
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of phi(x^2) * phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/24) * eta(q^3)^2 * eta(q^4)^5 / (eta(q) * eta(q^2) * eta(q^6) * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ 1, 2, -1, -3, 1, 1, 1, -1, -1, 2, 1, -4, 1, 2, -1, -1, 1, 1, 1, -3, -1, 2, 1, -2, ...].
a(25*n + 1) = a(n). a(25*n + 6) = a(25*n + 11) = a(25*n + 16) = a(25*n + 21) = 0.
Expansion of phi(-x^3) * f(x^2)^2 / psi(-x) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Nov 06 2015
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EXAMPLE
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G.f. = 1 + x + 3*x^2 + 2*x^3 + 2*x^4 + x^5 + 3*x^7 + 2*x^8 + 4*x^9 + 2*x^10 + ...
G.f. = q + q^25 + 3*q^49 + 2*q^73 + 2*q^97 + q^121 + 3*q^169 + 2*q^193 + 4*q^217 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 1}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &]]]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x^2] QPochhammer[ -x, x], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ -x, x^3] QPochhammer[ -x^2, x^3] QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 24*n + 1; sumdiv(n, d, kronecker( -12, d) * (n/d %2)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^6 + A) * eta(x^8 + A)^2), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A131963
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Expansion of f(x, x^2) * f(x^4, x^12) in powers of x where f(, ) is Ramanujan's general theta function.
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+0
11
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1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 2, 0, 0, 1, 0, 2, 1, 3, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 3, 0, 1, 0, 0, 1, 2, 2, 0, 1, 1, 2
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OFFSET
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0,6
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COMMENTS
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LINKS
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FORMULA
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Expansion of psi(x^4) * phi(-x^3) / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-13/24) * eta(q^2) * eta(q^3)^2 * eta(q^8)^2 / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 24 sequence [ 1, 0, -1, 1, 1, -1, 1, -1, -1, 0, 1, 0, 1, 0, -1, -1, 1, -1, 1, 1, -1, 0, 1, -2, ...].
a(25*n + 13) = a(n). a(25*n + 3) = a(25*n + 8) = a(25*n + 18) = a(25*n + 23) = 0.
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EXAMPLE
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G.f. = 1 + x + x^2 + x^4 + 2*x^5 + x^6 + x^7 + x^9 + x^11 + 2*x^12 + x^13 + ...
G.f. = q^13 + q^37 + q^61 + q^109 + 2*q^133 + q^157 + q^181 + q^229 + q^277 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 13}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 04 2015 *)
a[ n_] := SeriesCoefficient[(1/2) x^(-1/2) EllipticTheta[ 4, 0, x^3] QPochhammer[ -x, x] EllipticTheta[ 2, 0, x^2], {x, 0, n}]; (* Michael Somos, Nov 04 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 24*n + 13; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A131962
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Expansion of psi(x) * phi(-x^12) / chi(-x^4) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
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+0
10
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1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 3, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 0, 3, 1, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 2
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OFFSET
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0,11
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-7/24) * eta(q^2)^2 * eta(q^8) * eta(q^12)^2/( eta(q) * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, -2, ...].
a(25*n + 7) = a(n). a(25*n + 2) = a(25*n + 12) = a(25*n + 17) = a(25*n + 22) = 0.
Expansion of chi(x) * f(-x^8) * phi(-x^12) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Nov 06 2015
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EXAMPLE
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G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^7 + q^31 + q^79 + q^103 + q^127 + q^151 + q^175 + q^199 + q^223 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 7}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] EllipticTheta[ 4, 0, x^12] QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x^12] QPochhammer[ -x^4, x^4], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 24*n + 7; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^24 + A)), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A131964
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Expansion of f(x^2, x^10) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
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+0
8
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1, 1, 1, 2, 0, 1, 1, 0, 1, 0, 2, 1, 1, 1, 0, 1, 2, 2, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 1, 1, 3, 0, 0, 0, 2, 1, 1, 2, 1, 2, 1, 0, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 2, 1, 2, 0, 2, 2, 1, 1, 0, 0, 1
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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Expansion of phi(-x^4) * psi(-x^6) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-19/24) * eta(q^2) * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 1, 0, 1, -2, 1, -1, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, ...].
a(25*n + 19) = a(n). a(25*n + 4) = a(25*n + 9) = a(25*n + 14) = a(25*n + 24) = 0.
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EXAMPLE
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G.f. = 1 + x + x^2 + 2*x^3 + x^5 + x^6 + x^8 + 2*x^10 + x^11 + x^12 + x^13 + ...
G.f. = q^19 + q^43 + q^67 + 2*q^91 + q^139 + q^163 + q^211 + 2*q^259 + q^283 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 19}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) EllipticTheta[ 4, 0, x^4] QPochhammer[ -x, x] EllipticTheta[ 2, Pi/4, x^3], {x, 0, n}]; (* Michael Somos, Nov 03 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 24*n + 19; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A136748
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Expansion of (a(q) - a(q^2) - 4*a(q^4) + 4*a(q^8)) / 6 in powers of q where a() is a cubic AGM theta function.
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+0
3
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1, -1, 1, -3, 0, -1, 2, 3, 1, 0, 0, -3, 2, -2, 0, -3, 0, -1, 2, 0, 2, 0, 0, 3, 1, -2, 1, -6, 0, 0, 2, 3, 0, 0, 0, -3, 2, -2, 2, 0, 0, -2, 2, 0, 0, 0, 0, -3, 3, -1, 0, -6, 0, -1, 0, 6, 2, 0, 0, 0, 2, -2, 2, -3, 0, 0, 2, 0, 0, 0, 0, 3, 2, -2, 1, -6, 0, -2, 2, 0
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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Expansion of eta(q) * eta(q^3) * eta(q^4)^4 * eta(q^24)^2 / (eta(q^2) * eta(q^8) * eta(q^12))^2 in powers of q.
Euler transform of period 24 sequence [ -1, 1, -2, -3, -1, 0, -1, -1, -2, 1, -1, -2, -1, 1, -2, -1, -1, 0, -1, -3, -2, 1, -1, -2, ...].
a(n) is multiplicative with a(2) = -1, a(2^e) = -3 * (-1)^e if e>1, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123484.
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^2 * (1 + x^k + x^(2*k)) * (1 - x^(4*k) + x^(8*k))^2.
Moebius transform is period 24 sequence [ 1, -2, 0, -2, -1, 0, 1, 6, 2, -1, 0, 1, -2, 0, -6, -1, 0, 1, 2, 0, 2, -1, 0, ...].
a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
Expansion of q * f(-q, -q) * f(q^2, q^10) / f(-q, -q^5)^2 in powers of q where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 12 2015
Sum_{k=1..n} abs(a(k)) ~ (Pi*sqrt(3)/4) * n. - Amiram Eldar, Jan 28 2024
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EXAMPLE
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G.f. = q - q^2 + q^3 - 3*q^4 - q^6 + 2*q^7 + 3*q^8 + q^9 - 3*q^12 + 2*q^13 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, DivisorSum[ n, (Mod[#, 2] - 4 Boole[Mod[#, 8] == 4]) KroneckerSymbol[ -3, n/#] &]]; (* Michael Somos, Oct 12 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[# == 1 || # == 3, 1, # == 2, If[#2 < 2, -1, -3 (-1)^#2], Mod[#, 6] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* Michael Somos, Oct 12 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3+A) * eta(x^4 + A)^4 * eta(x^24 + A)^2 / ( eta(x^2 + A) * eta(x^8+A) * eta(x^12+A) )^2, n))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, if( e<2, -1, -3 * (-1)^e), p==3, 1, p%6>1, !(e%2), e+1)))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, ((d%2) -4 * (d%8==4)) * kronecker(-3, n/d)))};
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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