A304629 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(5*k)))^n.
1, 1, 3, 13, 51, 201, 819, 3389, 14131, 59341, 250703, 1064207, 4535091, 19390229, 83139955, 357354213, 1539272499, 6642769925, 28714955571, 124312591469, 538895612751, 2338948779320, 10162837993377, 44202371860240, 192431323820851, 838442649862701, 3656031108325651
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..500
Programs
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Mathematica
Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(5 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}] Table[SeriesCoefficient[Product[1/(1 - x^k + x^(2 k) - x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}] (* Calculation of constants {d,c}: *) With[{k = 5}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s] / QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)
Formula
a(n) = [x^n] Product_{k>=1} 1/(1 - x^k + x^(2*k) - x^(3*k) + x^(4*k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.445766346387064439086120427... and c = 0.267035948020079842478289... - Vaclav Kotesovec, May 18 2018