A296163 a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.
1, 1, 5, 22, 105, 501, 2456, 12160, 60801, 306130, 1550255, 7887034, 40281720, 206405967, 1060602800, 5463059772, 28199365873, 145832364580, 755420838614, 3918935839970, 20357605331355, 105878815699042, 551273881133750, 2873161931172668, 14988243880188600
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Partition Function b_k
Programs
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Mathematica
Table[SeriesCoefficient[Product[((1 - x^(5 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}] Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k) + x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}] (* Calculation of constant d: *) With[{k = 5}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)
Formula
a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k) + x^(4*k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 5.3271035802753567624196808294779171420899175782347488197... and c = 0.2712048688090020853684153670711011713396954... - Vaclav Kotesovec, May 13 2018
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