A261369 Expansion of (psi(-x^3) / f(x))^2 in powers of x where psi(), f() are Ramanujan theta functions.

1, -2, 5, -12, 24, -46, 86, -152, 262, -442, 725, -1168, 1852, -2886, 4436, -6736, 10103, -14994, 22040, -32092, 46336, -66380, 94378, -133256, 186926, -260576, 361126, -497716, 682340, -930774, 1263624, -1707672, 2297737, -3078850, 4109022, -5462924, 7236280

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..2000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2)^3 * eta(q^6)))^2 in powers of q.

Euler transform of period 12 sequence [ -2, 4, -4, 2, -2, 4, -2, 2, -4, 4, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A262930. - Michael Somos, Nov 07 2015

a(n) = A187153(3*n + 2) = A213265(3*n + 2) = A262930(3*n + 2). - Michael Somos, Nov 07 2015

Convolution square of A139135. - Michael Somos, Nov 07 2015

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (8*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 - 2*x + 5*x^2 - 12*x^3 + 24*x^4 - 46*x^5 + 86*x^6 - 152*x^7 + ...
G.f. = q^2 - 2*q^5 + 5*q^8 - 12*q^11 + 24*q^14 - 46*q^17 + 86*q^20 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^3 * eta(x^6 + A)))^2, n))};

Crossrefs

Cf. A139135, A187153, A213265, A262930.

Sequence in context: A294868 A192981 A116713 * A116714 A023615 A320235

Adjacent sequences: A261366 A261367 A261368 * A261370 A261371 A261372

Keyword

sign

Author

Michael Somos, Aug 16 2015

Status

approved