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a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.
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%I #10 Feb 16 2025 08:33:52

%S 1,1,5,22,105,501,2456,12160,60801,306130,1550255,7887034,40281720,

%T 206405967,1060602800,5463059772,28199365873,145832364580,

%U 755420838614,3918935839970,20357605331355,105878815699042,551273881133750,2873161931172668,14988243880188600

%N a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionb.html">Partition Function b_k</a>

%F a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k) + x^(4*k))^n.

%F a(n) ~ c * d^n / sqrt(n), where d = 5.3271035802753567624196808294779171420899175782347488197... and c = 0.2712048688090020853684153670711011713396954... - _Vaclav Kotesovec_, May 13 2018

%t Table[SeriesCoefficient[Product[((1 - x^(5 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]

%t 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}]

%t (* 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 *)

%Y Cf. A008485, A035959, A121591, A263002, A270913, A285928, A296044, A296162.

%K nonn,changed

%O 0,3

%A _Ilya Gutkovskiy_, Dec 06 2017