A262146 - OEIS

A262146

Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 4, 8, 15, 25, 42, 68, 107, 166, 253, 377, 557, 811, 1166, 1661, 2344, 3275, 4543, 6253, 8544, 11600, 15653, 20994, 28011, 37178, 49100, 64550, 84489, 110115, 142951, 184867, 238196, 305844, 391391, 499244, 634865, 804925, 1017610, 1282957, 1613195

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

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

a(n) = A132217(2*n + 1) = - A262160(2*n + 1).

Convolution product of A097451 and A078408.

a(n) ~ exp(Pi*sqrt(n)) / (2^(7/2) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Mar 31 2018

EXAMPLE

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 25*x^5 + 42*x^6 + 68*x^7 + ...

G.f. = q^13 + 2*q^29 + 4*q^45 + 8*q^61 + 15*q^77 + 25*q^93 + 42*q^109 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ - x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n + 1}];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( - eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};

CROSSREFS

Cf. A078408, A097451, A132217, A262160.