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Search: a123649 -id:a123649
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A187153
Expansion of q * (psi(q) / psi(q^2)) / (psi(q^3) / psi(q^6))^3 in powers of q where psi() is a Ramanujan theta function.
+10
5
1, 1, -1, -3, -2, 3, 8, 5, -7, -18, -12, 15, 38, 24, -30, -75, -46, 57, 140, 86, -104, -252, -152, 183, 439, 262, -313, -744, -442, 522, 1232, 725, -852, -1998, -1168, 1365, 3182, 1852, -2150, -4986, -2886, 3336, 7700, 4436, -5106, -11736, -6736, 7719
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
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).
The Alaca, et. al. paper in equation (1.8) on page 178 uses the notation p(q) := (phi^2(q) - phi^2(q^3)/(2 phi^2(q^3)), where phi(q) is a Ramanujan theta function. This p(q) is twice the g.f. of this sequence. - Michael Somos, Mar 30 2021
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
A. Alaca, S. Alaca, K. S. Williams, On the two-dimensional theta functions of the Borweins
Johannes Blümlein, Iterative Non-iterative Integrals in Quantum Field Theory, arXiv:1808.08128 [hep-th], 2018.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
W. Zudilin, Deciding whether a given power series is modular or not
FORMULA
Expansion of (eta(q^2) * eta(q^3) * eta(q^12)^2)^3 / (eta(q) * eta(q^4)^2 * eta(q^6)^9) in powers of q.
Euler transform of period 12 sequence [1, -2, -2, 0, 1, 4, 1, 0, -2, -2, 1, 0, ...].
Expansion of c(q) * c(q^4)^2 / c(q^2)^3 in powers of q where c() is a cubic AGM theta function.
If p = 2 * A(q), then B(q) = p * ((2 + p) / (1 + 2*p))^3 and B(q^3) = p^3 * ((2 + p) / (1 + 2*p)) where B() is the g.f. for A115977. - Michael Somos, Feb 27 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u * (1 + 2*v))^2 - v * (1 + 2*u).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^2 * (1 + 4*v) * (3*v + u*(1 + 4*v)) - v * (1 + v) * (3*u + 1 + v).
Convolution inverse of A187143.
Expansion of (phi^2(q) - phi^2(q^3)/(4*phi^2(q^3)), where phi(q) is a Ramanujan theta function. - Michael Somos, Mar 30 2021
EXAMPLE
G.f. = q + q^2 - q^3 - 3*q^4 - 2*q^5 + 3*q^6 + 8*q^7 + 5*q^8 - 7*q^9 - 18*q^10 + ...
MATHEMATICA
QP = QPochhammer; s = (QP[q^2]*QP[q^3]*QP[q^12]^2)^3/(QP[q]*QP[q^4]^2* QP[q^6]^9) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
QP = QPochhammer; Rest[Table[SeriesCoefficient[q*(QP[-q, q^2]*QP[-q^6, q^6]^3)/(QP[-q^2, q^2]*QP[-q^3, q^6]^3), {q, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Dec 04 2017 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^2, x^4]^2 QPochhammer[x^3, x^6]^3 QPochhammer[ -x^6, x^6]^6, {x, 0, n}]; (* Michael Somos, Oct 15 2018 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, x]^2 / EllipticTheta[3, 0, x^3]^2 - 1)/4, {x, 0, n}]; (* Michael Somos, Mar 30 2021 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^2)^3 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^9), n))};
CROSSREFS
Cf. A115977, A123649, A187143, A213265.
KEYWORD
sign
AUTHOR
Michael Somos, Mar 06 2011
STATUS
approved
A217771
Expansion of (phi(-x) / phi(-x^3))^2 in powers of x where phi() is a Ramanujan theta function.
+10
4
1, -4, 4, 4, -12, 8, 12, -32, 20, 28, -72, 48, 60, -152, 96, 120, -300, 184, 228, -560, 344, 416, -1008, 608, 732, -1756, 1048, 1252, -2976, 1768, 2088, -4928, 2900, 3408, -7992, 4672, 5460, -12728, 7408, 8600, -19944, 11544, 13344, -30800, 17744, 20424
(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
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q)^4 * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)^4) in powers of q.
Euler transform of period 6 sequence [ -4, -2, 0, -2, -4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u) * (u + v^2) - 4 * u.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (3 + u * v)^2 - v * (3*u + v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A217786.
a(n) = - 4 * A123649(n) unless n=0.
Convolution inverse of A186924. Convolution square of A139137.
EXAMPLE
G.f. = 1 - 4*x + 4*x^2 + 4*x^3 - 12*x^4 + 8*x^5 + 12*x^6 - 32*x^7 + 20*x^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}]; (* Michael Somos, Mar 24 2013 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^4), n))}
CROSSREFS
Cf. A123649, A186924, A217786.
KEYWORD
sign
AUTHOR
Michael Somos, Mar 24 2013
STATUS
approved
A187144
McKay-Thompson series of class 12I for the Monster group with a(0) = 1.
+10
1
1, 1, 2, 0, 1, 0, 0, 0, -2, 0, -2, 0, 2, 0, 4, 0, 3, 0, -4, 0, -8, 0, -4, 0, 5, 0, 14, 0, 7, 0, -8, 0, -20, 0, -12, 0, 14, 0, 28, 0, 17, 0, -20, 0, -44, 0, -24, 0, 28, 0, 66, 0, 36, 0, -40, 0, -90, 0, -52, 0, 56, 0, 124, 0, 71, 0, -80, 0, -176, 0, -96, 0, 109, 0, 244, 0, 133, 0, -144
(list; graph; refs; listen; history; text; internal format)
OFFSET
-1,3
COMMENTS
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
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
FORMULA
Expansion of c(q) / c(q^4) in powers of q where c() is a cubic AGM function.
Expansion of eta(q^3)^3 * eta(q^4) / (eta(q) * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ 1, 1, -2, 0, 1, -2, 1, 0, -2, 1, 1, 0, ...].
Convolution inverse of A123649.
a(2*n) = 0 unless n=0. a(2*n - 1) = A058487(n).
EXAMPLE
G.f. = 1/q + 1 + 2*q + q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...
MATHEMATICA
QP = QPochhammer; s = QP[q^3]^3*(QP[q^4]/(QP[q]*QP[q^12]^3)) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) / (eta(x + A) * eta(x^12 + A)^3), n))};
CROSSREFS
Cf. A058487, A123649.
KEYWORD
sign
AUTHOR
Michael Somos, Mar 05 2011
STATUS
approved
A261446
Expansion of f(-x^3, -x^3) * f(-x, -x^5) / f(-x, -x)^2 in powers of x where f(,) is Ramanujan's general theta function.
+10
1
1, 3, 8, 18, 38, 75, 140, 252, 439, 744, 1232, 1998, 3182, 4986, 7700, 11736, 17673, 26322, 38808, 56682, 82070, 117867, 167996, 237744, 334202, 466836, 648224, 895014, 1229148, 1679436, 2283568, 3090672, 4164578, 5587941, 7467464, 9940482, 13183238, 17421288
(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 f(-x^2) * f(-x^3) * f(-x^6) / f(-x)^3 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/3) * eta(q^2) * eta(q^3) * eta(q^6) / eta(q)^3 in powers of q.
Euler transform of period 6 sequence [ 3, 2, 2, 2, 3, 0, ...].
a(n) = (-1)^n * A261325(n). 2 * a(2*n) = A261240(3*n + 1). a(2*n + 1) = 3 * A233698(n).
2 * a(n) = A058647(3*n + 1) = A139213(3*n + 1) = A186964(3*n + 1) = A187020(3*n + 1).
a(n) = A123649(3*n + 1) = A139214(3*n + 1) = A233693(3*n + 1).
Convolution inverse is A132301.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
EXAMPLE
G.f. = 1 + 3*x + 8*x^2 + 18*x^3 + 38*x^4 + 75*x^5 + 140*x^6 + 252*x^7 + ...
G.f. = q + 3*q^4 + 8*q^7 + 18*q^10 + 38*q^13 + 75*q^16 + 140*q^19 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] QPochhammer[ x^6] / QPochhammer[ x]^3, {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(3*k)) * (1-x^(6*k)) / (1-x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A)^3, n))};
CROSSREFS
Cf. A058647, A123649, A132301, A139213, A139214, A186964, A187020, A233693, A233698, A261240, A261325.
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 19 2015
STATUS
approved
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