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

Author

Michael Somos, Sep 03 2015

Keywords

• nonn

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).

Examples

			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 + ...
		

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

Crossrefs

Cf. A122135, A147699.

Programs

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}];

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))};

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.