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Search: a132973 -id:a132973
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A205970
a(n) = Fibonacci(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.
+20
5
1, -3, 3, -6, 9, 0, 24, -78, 63, -102, 0, 0, 432, -1398, 2262, 0, 2961, 0, 7752, -25086, 0, -65676, 0, 0, 139104, -225075, 728358, -589254, 1906866, 0, 0, -8077614, 6534927, 0, 0, 0, 44791056, -144946902, 234529014, -379475916, 0, 0, 1607485776, -2600966622, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Compare g.f. to the Lambert series of A132973:
1 - 3*Sum_{n>=0} x^(6*n+1)/(1+x^(6*n+1)) - x^(6*n+5)/(1+x^(6*n+5)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1 - 3*Sum_{n>=0} Fibonacci(6*n+1)*x^(6*n+1)/(1 + Lucas(6*n+1) * x^(6*n+1) - x^(12*n+2)) - Fibonacci(6*n+5)*x^(6*n +5)/(1 + Lucas(6*n+5) * x^(6*n+5) - x^(12*n+10)).
EXAMPLE
G.f.: A(x) = 1 - 3*x + 3*x^2 - 6*x^3 + 9*x^4 + 24*x^6 - 78*x^7 + 63*x^8 +...
where A(x) = 1 - 1*3*x + 1*3*x^2 - 2*3*x^3 + 3*3*x^4 + 8*3*x^6 - 13*6*x^7 + 21*3*x^8 +...+ Fibonacci(n)*A132973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 3*( 1*x/(1+x-x^2) - 5*x^5/(1+11*x^5-x^10) + 13*x^7/(1+29*x^7-x^14) - 89*x^11/(1+199*x^11-x^22) + 233*x^13/(1+521*x^13-x^26) - 1597*x^17/(1+3571*x^17-x^34) +...).
MATHEMATICA
A132973:= CoefficientList[Series[(-1)^(-1/4)*EllipticTheta[2, 0, I*Sqrt[q]]^3/EllipticTheta[2, 0, I*Sqrt[q^3]]/4, {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n]*A132973[[n + 1]]], {n, 0, 50}] (* G. C. Greubel, Dec 03 2017 *)
PROG
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 3*sum(m=0, n, fibonacci(6*m+1)*x^(6*m+1)/(1+Lucas(6*m+1)*x^(6*m+1)-x^(12*m+2) +x*O(x^n)) - fibonacci(6*m+5)*x^(6*m+5)/(1+Lucas(6*m+5)*x^(6*m+5)-x^(12*m+10) +x*O(x^n)) ), n)}
for(n=0, 61, print1(a(n), ", "))
CROSSREFS
Cf. A132973, A205969, A205971, A203847.
Cf. A209450 (Pell variant).
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 04 2012
STATUS
approved
A209450
a(n) = Pell(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.
+20
4
1, -3, 6, -15, 36, 0, 210, -1014, 1224, -2955, 0, 0, 41580, -200766, 484692, 0, 1412496, 0, 8232630, -39750654, 0, -231683790, 0, 0, 1630019160, -3935214363, 19000895772, -22936110135, 110745336312, 0, 0, -1558305137094, 1881040698144, 0, 0, 0, 63900011068740
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Compare g.f. to the Lambert series of A132973: 1 - 3*Sum_{n>=0} x^(6*n+1)/(1+x^(6*n+1)) - x^(6*n+5)/(1+x^(6*n+5)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1 - 3*Sum_{n>=0} Pell(6*n+1)*x^(6*n+1)/(1+A002203(6*n+1)*x^(6*n+1)-x^(12*n+2)) - Pell(6*n+5)*x^(6*n+5)/(1+A002203(6*n+5)*x^(6*n+5)-x^(12*n+10)), where A002203(n) = Pell(n-1) + Pell(n+1).
EXAMPLE
G.f.: A(x) = 1 - 3*x + 6*x^2 - 15*x^3 + 36*x^4 + 210*x^6 - 1014*x^7 +...
where A(x) = 1 - 1*3*x + 2*3*x^2 - 5*3*x^3 + 12*3*x^4 + 70*3*x^6 - 169*6*x^7 + 408*3*x^8 +...+ Pell(n)*A132973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 3*( 1*x/(1+2*x-x^2) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 5741*x^11/(1+16238*x^11-x^22) + 33461*x^13/(1+94642*x^13-x^26) - 1136689*x^17/(1+3215042*x^17-x^34) +...).
MATHEMATICA
A132973[n_]:= SeriesCoefficient[EllipticTheta[2, Pi/4, q^(1/2)]^3/EllipticTheta[2, Pi/4, q^(3/2)]/2, {q, 0, n}]; Join[{1}, Table[ Fibonacci[n, 2]*A132973[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)
PROG
(PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 - 3*sum(m=0, n, Pell(6*m+1)*x^(6*m+1)/(1+A002203(6*m+1)*x^(6*m+1)-x^(12*m+2) +x*O(x^n)) - Pell(6*m+5)*x^(6*m+5)/(1+A002203(6*m+5)*x^(6*m+5)-x^(12*m+10) +x*O(x^n)) ), n)}
for(n=0, 61, print1(a(n), ", "))
CROSSREFS
Cf. A132973, A205970, A209449, A209451, A204270, A000129 (Pell), A002203.
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 10 2012
STATUS
approved
A107760
Expansion of eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2) in powers of q.
+10
8
1, 3, 3, 3, 3, 0, 3, 6, 3, 3, 0, 0, 3, 6, 6, 0, 3, 0, 3, 6, 0, 6, 0, 0, 3, 3, 6, 3, 6, 0, 0, 6, 3, 0, 0, 0, 3, 6, 6, 6, 0, 0, 6, 6, 0, 0, 0, 0, 3, 9, 3, 0, 6, 0, 3, 0, 6, 6, 0, 0, 0, 6, 6, 6, 3, 0, 0, 6, 0, 0, 0, 0, 3, 6, 6, 3, 6, 0, 6, 6, 0, 3, 0, 0, 6, 0, 6, 0, 0, 0, 0, 12, 0, 6, 0, 0, 3, 6, 9, 0, 3, 0, 0, 6, 6
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
REFERENCES
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.42).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u^2*w + 4 * v*w^2 - 4 * v^2*w - 2 * u*v*w.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 - u2) * (u1 - u2 - u3 + u6) - 3 * u6 * (u2 - u6).
Expansion of psi(q)^3 / psi(q^3) in powers of q where psi() is a Ramanujan theta function.
Expansion of (a(q) + a(q^2)) / 2 = b(q^2)^2 / b(q) in powers of q where a(), b() are cubic AGM theta functions. - Michael Somos, Aug 30 2008
Euler transform of period 6 sequence [ 3, -3, 2, -3, 3, -2, ...].
Moebius transform is period 6 sequence [ 3, 0, 0, 0, -3, 0, ...]. - Michael Somos, Aug 11 2009
a(n) = 3 * b(n) unless n=0 and b() is multiplicative with b(p^e) = 1 if p=2 or p=3; b(p^e) = 1+e if p == 1 (mod 6); b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6). - Michael Somos, Aug 11 2009
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123330. - Michael Somos, Aug 11 2009
G.f.: (Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1)))^3 / (Product_{k>0} (1 - x^(6*k)) / (1 - x^(6*k - 3))). - Michael Somos, Aug 11 2009
a(n) = 3 * A035178(n) unless n=0. a(n) = (-1)^n * A132973. a(2*n) = a(3*n) = a(n). a(6*n + 5) = 0. a(2*n + 1) = 3 * A033762. a(3*n + 1) = 3 * A033687(n). a(4*n + 1) = 3 * A112604(n). a(4*n + 3) = 3 * A112605(n). a(6*n + 1) = 3 * A097195(n). Convolution inverse of A132979.
a(8*n + 1) = 3 * A112606(n). a(8*n + 3) = 3* A112608(n). a(8*n + 5) = 6 * A112607(n-1). a(8*n + 7) = 6 * A112609(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3)/2 = 2.720699... . - Amiram Eldar, Dec 28 2023
EXAMPLE
G.f. = 1 + 3*q + 3*q^2 + 3*q^3 + 3*q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + ...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], 3 Times @@ (Which[ # < 5, 1, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1 ] & @@@ FactorInteger@n)]; (* Michael Somos, Aug 04 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^3 / (4 EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Aug 04 2015 *)
QP = QPochhammer; s = QP[q^3]*(QP[q^2]^6/(QP[q]^3*QP[q^6]^2)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 3 * direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -12, p) * X)))[n])};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^2 + A)^6 / (eta(x^6 + A)^2 * eta(x + A)^3), n))};
(PARI) {a(n) = if ( n<1, n==0, 3 * sumdiv( n, d, kronecker( -12, d)))};
(Sage) A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 3*A[1] # Michael Somos, Sep 27 2013
(Magma) A := Basis( ModularForms( Gamma1(6), 1), 88); A[1] + 3*A[2]; /* Michael Somos, Aug 04 2015 */
CROSSREFS
Cf. A033687, A033762, A035178, A097195, A112604, A112605, A123330, A132973, A132979.
Cf. A112606, A112607, A112608, A112609. - Michael Somos, Sep 27 2013
Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.
KEYWORD
nonn,easy
AUTHOR
Michael Somos, May 24 2005
STATUS
approved
A132974
Expansion of psi(-q^3) / psi(-q)^3 in powers of q where psi() is a Ramanujan theta function.
+10
8
1, 3, 6, 12, 24, 45, 78, 132, 222, 363, 576, 900, 1392, 2121, 3180, 4716, 6936, 10098, 14550, 20796, 29520, 41595, 58176, 80856, 111750, 153561, 209820, 285240, 385968, 519840, 696960, 930516, 1237470, 1639314, 2163456, 2845080, 3728904, 4871211
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2)^3 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6) ) in powers of q.
Euler transform of period 12 sequence [3, 0, 2, 3, 3, 0, 3, 3, 2, 0, 3, 2, ...].
G.f.: Product_{k>0} (1 - x^(3*k)) * (1 + x^(6*k)) / ( (1 - x^k) * (1 + x^(2*k)) )^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (108)^(-1/2) (t/i)^(-1) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A133637.
A132979(n) = (-1)^n * a(n). Convolution inverse of A132973.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(5/4)). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = 1 + 3*q + 6*q^2 + 12*q^3 + 24*q^4 + 45*q^5 + 78*q^6 + 132*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(3/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)]^3 , {q, 0, n}]; (* Michael Somos, Sep 26 2017 *)
nmax=60; CoefficientList[Series[Product[(1-x^(3*k)) * (1+x^(6*k)) / ( (1-x^k)^3 * (1+x^(2*k))^3 ), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A ) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};
CROSSREFS
Cf. A132973, A132979.
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 07 2007
STATUS
approved
A113447
Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.
+10
7
1, 1, 1, -1, 0, 1, 2, 1, 1, 0, 0, -1, 2, 2, 0, -1, 0, 1, 2, 0, 2, 0, 0, 1, 1, 2, 1, -2, 0, 0, 2, 1, 0, 0, 0, -1, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, -1, 3, 1, 0, -2, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, -1, 0, 0, 2, 0, 0, 0, 0, 1, 2, 2, 1, -2, 0, 2, 2, 0, 1, 0, 0, -2, 0, 2, 0, 0, 0, 0, 4, 0, 2, 0, 0, 1, 2, 3, 0, -1, 0, 0, 2, 2, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,7
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Li-Chien Shen, On the Modular Equations of Degree 3, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1105, equation (3.8).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of (eta(q^2) * eta(q^3)^3 * eta(q^12)^3) / (eta(q) * eta(q^4) * eta(q^6)^3) in powers of q.
Euler transform of period 12 sequence [1, 0, -2, 1, 1, 0, 1, 1, -2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0, ...].
a(n) is multiplicative and a(2^e) = -(-1)^e if e>0, 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.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - 2 * x^(12*k - 8) / (1 - x^(12*k - 8)) + 2 * x^(12*k - 4) / (1 - x^(12*k-4)).
G.f.: Sum_{k>0} x^k * (1 - x^(3*k))^2 / (1 + x^(4*k) + x^(8*k)).
G.f.: x * Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)) * ((1 - x^(12*k - 6)) / (1 - x^(3*k)))^3.
Expansion of theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.
Expansion of q * psi(-q^3)^3 / psi(-q) in powers of q where psi() is a Ramanujan theta function.
Expansion of (c(q) * c(q^4)) / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132973.
a(n) = -(-1)^n * A093829(n). - Michael Somos, Jan 31 2015
Convolution inverse of A133637.
a(3*n) = a(n). a(6*n + 5) = a(12*n + 10) = 0. |a(n)| = A035178(n).
a(2*n) = A093829(n). a(2*n + 1) = A033762(n).
a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n).
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 6) = A112605(n). a(8*n + 7) = 2 * A112609(n).
a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).
a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A121963(n). a(24*n + 19) = 2 * A131964(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 23 2023
EXAMPLE
G.f. = q + q^2 + q^3 - q^4 + q^6 + 2*q^7 + q^8 + q^9 - q^12 + 2*q^13 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, Jan 31 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, -(-1)^max( 1, valuation( n, 2)) * sumdiv(n, d, kronecker( -12, d)))};
(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, if( p==2, 1 + X / (1 + X), 1 / ((1 - X) * (1 - kronecker( -12, p) * X))))[n])};
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3), n))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1][d%12 + 1]))}; /* Michael Somos, May 07 2015 */
(Magma) A := Basis( ModularForms( Gamma1(24), 1), 106); A[2] + A[3] + A[4] - A[5] + A[7] + 2*A[8] + A[9] + A[10]; /* Michael Somos, May 07 2015 */
CROSSREFS
Cf. A033687, A033762, A035178, A073010, A093829, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964. A133637.
Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.
KEYWORD
sign,easy,mult
AUTHOR
Michael Somos, Nov 02 2005
STATUS
approved
A137608
Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 in powers of q where psi() is a Ramanujan theta function.
+10
3
1, -1, 1, -1, 0, -1, 2, -1, 1, 0, 0, -1, 2, -2, 0, -1, 0, -1, 2, 0, 2, 0, 0, -1, 1, -2, 1, -2, 0, 0, 2, -1, 0, 0, 0, -1, 2, -2, 2, 0, 0, -2, 2, 0, 0, 0, 0, -1, 3, -1, 0, -2, 0, -1, 0, -2, 2, 0, 0, 0, 2, -2, 2, -1, 0, 0, 2, 0, 0, 0, 0, -1, 2, -2, 1, -2, 0, -2, 2, 0, 1, 0, 0, -2, 0, -2, 0, 0, 0, 0, 4, 0, 2, 0, 0, -1, 2, -3, 0, -1, 0, 0, 2, -2, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,7
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (1 - b(q^2)^2 / b(-q) ) / 3 in powers of q where b() is a cubic AGM function.
Moebius transform is period 12 sequence [ 1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 unless e=0, 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.: Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
a(n) = -(-1)^n * A035178(n). -3 * a(n) = A132973(n) unless n = 0.
a(2*n) = -A035178(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A227696(n).
a(4*n + 1) + A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n-1). a(8*n + 7) = 2 * A112609(n).
a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).
a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A131963(n). a(24*n + 19) = 2 * A131964(n).
EXAMPLE
G.f. = q - q^2 + q^3 - q^4 - q^6 + 2*q^7 - q^8 + q^9 - q^12 + 2*q^13 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[n, KroneckerSymbol[ -12, #] &]]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ (4 + EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]) / 6, {q, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, May 07 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker(-12, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A))) / 3, n))}; /* Michael Somos, May 06 2015 */
CROSSREFS
Cf. A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964, A132973, A227696.
Cf. A035178, A093829, A113447 are same up to sign.
KEYWORD
sign,mult
AUTHOR
Michael Somos, Jan 29 2008
STATUS
approved
page 1

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