A262987 Expansion of f(-x, -x^5) * f(x^3, x^5) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, 1, 3, 6, 11, 19, 33, 53, 86, 134, 205, 309, 460, 672, 974, 1394, 1975, 2773, 3863, 5333, 7316, 9964, 13484, 18140, 24269, 32288, 42751, 56331, 73888, 96503, 125529, 162635, 209939, 270027, 346123, 442213, 563205, 715110, 905361, 1142998, 1439098, 1807175

Offset: 0

Author

Michael Somos, Oct 06 2015

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 33*x^6 + 53*x^7 + ...
G.f. = q^5 + q^21 + 3*q^37 + 6*q^53 + 11*q^69 + 19*q^85 + 33*q^101 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A035294, A097451, A132217, A262160.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n}];
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x, -x^5]*f[x^3, x^5]/f[-x, -x^2]^2, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 31 2018 *)

PARI

{a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};

Formula

Expansion of (psi(x^6) / psi(x) + psi(x^6) / psi(-x)) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

Euler transform of period 48 sequence [1, 2, 3, 2, 2, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 0, 1, 1, 3, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 2, 2, 3, 2, 1, 0, ...].

a(n) = A132217(2*n) = A262160(2*n).

Convolution product of A035294 and A097451.

a(n) ~ exp(sqrt(n)*Pi)/(8*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2015