A258593 Expansion of (phi(x^2) * psi(x^2) / phi(-x)^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 8, 46, 208, 805, 2776, 8742, 25584, 70450, 184232, 460832, 1108848, 2578295, 5814992, 12760598, 27317056, 57174768, 117223008, 235818894, 466154416, 906606234, 1736736024, 3280271526, 6114139616, 11255369609, 20478505104, 36849912318, 65619691088

Offset

0,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (f(x^2)^2 / (chi(-x^2) * phi(-x)^2))^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Expansion of q^(-1/2) * (eta(q^4)^7 / (eta(q)^4 * eta(q^2) * eta(q^8)^2))^2 in powers of q.

Euler transform of period 8 sequence [ 8, 10, 8, -4, 8, 10, 8, 0, ...].

-4 * a(n) = A260186(2*n + 1).

a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Example

G.f. = 1 + 8*x + 46*x^2 + 208*x^3 + 805*x^4 + 2776*x^5 + 8742*x^6 + ...
G.f. = q + 8*q^3 + 46*q^5 + 208*q^7 + 805*q^9 + 2776*q^11 + 8742*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) (EllipticTheta[ 3, 0, x^2] EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0 , x]^2)^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2]^2 QPochhammer[ -x^2, x^2] / EllipticTheta[ 4, 0, x]^2)^2, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^7 / (eta(x + A)^4 * eta(x^2 + A) * eta(x^8 + A)^2))^2, n))};

Crossrefs

Cf. A260186.

Sequence in context: A034469 A212673 A183392 * A134114 A071586 A027650

Adjacent sequences: A258590 A258591 A258592 * A258594 A258595 A258596

Keyword

nonn

Author

Michael Somos, Nov 06 2015

Status

approved