A327716 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003823.

1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 7, 9, 10, 12, 14, 17, 21, 23, 26, 32, 40, 45, 51, 58, 69, 80, 89, 102, 116, 135, 154, 177, 198, 224, 253, 288, 326, 361, 408, 459, 521, 583, 650, 723, 812, 909, 1009, 1122, 1244, 1393, 1547, 1716, 1898, 2101, 2326, 2575, 2845, 3132, 3456, 3809

Offset

0,5

Comments

a(n) > 0.

Links

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

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Formula

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(5*j-2))) * (1-x^(i*(5*j-3))) / ((1-x^(i*(5*j-1))) * (1-x^(i*(5*j-4)))).

G.f.: Product_{k>=1} (1-x^k)^(-A035187(k)).

a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 4*log((1+sqrt(5))/2) / (3*sqrt(5)) = 0.2869392939760026925..., c = 0.203427046022096... - Vaclav Kotesovec, Sep 24 2019, updated May 09 2020

Mathematica

nmax = 60; CoefficientList[Series[Product[QPochhammer[x^(5*j - 3)] * QPochhammer[x^(5*j - 2)]/(QPochhammer[x^(5*j - 4)] * QPochhammer[x^(5*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)

Prog

(PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(5, d))))

Crossrefs

Convolution inverse of A327688.

Cf. A003823, A035187, A327690, A327691, A327694, A327717, A327718, A327719, A327720.

Sequence in context: A029066 A174522 A327719 * A327720 A327718 A035581

Adjacent sequences: A327713 A327714 A327715 * A327717 A327718 A327719

Keyword

nonn

Author

Seiichi Manyama, Sep 23 2019

Status

approved