A261526 Expansion of (H(-x) / chi(-x))^2 in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function.

1, 2, 5, 8, 14, 22, 36, 54, 83, 120, 176, 250, 356, 494, 687, 936, 1276, 1714, 2298, 3046, 4030, 5280, 6902, 8952, 11580, 14882, 19077, 24314, 30910, 39104, 49344, 62000, 77712, 97032, 120872, 150058, 185869, 229520, 282814, 347504, 426118, 521182, 636204

Offset

0,2

Comments

Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 20 sequence [ 2, 2, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0, 2, 2, 0, ...].

Expansion of (f(x, -x^4) / f(-x^2, -x^2))^2 = (f(x^2, x^8) / f(-x, -x^4))^2 in powers of x where f(, ) is Ramanujan's general theta function.

G.f.: (Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1))))^2.

a(n) = - A147699(5*n + 2).

Convolution square of A122135.

Example

G.f. = 1 + 2*x + 5*x^2 + 8*x^3 + 14*x^4 + 22*x^5 + 36*x^6 + 54*x^7 + ...
G.f. = q^9 + 2*q^29 + 5*q^49 + 8*q^69 + 14*q^89 + 22*q^109 + 36*q^129 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^2, -x^5] QPochhammer[ -x^3, -x^5])^-2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{2, 2, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0, 2, 2, 0}[[Mod[k, 20, 1]]], {k, n}], {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 2, 2, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0, 2, 2][k%20 + 1]), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n + 1) - 1)\2, x^(k^2 + k)/ prod(i=1, 2*k+1, 1 - x^i, 1 + x * O(x^(n - k^2-k))))^2, n))};

Crossrefs

Cf. A122135, A147699.

Sequence in context: A327380 A330378 A058578 * A295392 A023674 A281864

Adjacent sequences: A261523 A261524 A261525 * A261527 A261528 A261529

Keyword

nonn

Author

Michael Somos, Sep 03 2015

Status

approved