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A327716 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003823. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
a(n) > 0.
LINKS
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.
Sequence in context: A029066 A174522 A327719 * A327720 A327718 A035581
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 23 2019
STATUS
approved

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Last modified July 19 22:34 EDT 2024. Contains 374441 sequences. (Running on oeis4.)