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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
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..42.

M. 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: A000094 A182377 A058578 * A295392 A023674 A281864

Adjacent sequences:  A261523 A261524 A261525 * A261527 A261528 A261529

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 03 2015

STATUS

approved

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Last modified April 24 04:19 EDT 2018. Contains 302984 sequences. (Running on oeis4.)