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%I #9 Jan 17 2024 08:14:04
%S 1,1,5,19,89,406,1913,9073,43505,209971,1019390,4971781,24343037,
%T 119579006,589050663,2908727839,14393759457,71360342129,354372852011,
%U 1762416036422,8776797353574,43761058841638,218431753457637,1091385769314272,5458068218285909,27319061357620056
%N a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionb.html">Partition Function b_k</a>
%F a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k))^n.
%F a(n) ~ c * d^n / sqrt(n), where d = 5.1069752682291604222843644516987970799026747758649349... and c = 0.271879273312907861082536692355942116774864... - _Vaclav Kotesovec_, May 13 2018
%t Table[SeriesCoefficient[Product[((1 - x^(3 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
%t Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
%t (* Calculation of constant d: *) With[{k = 3}, 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. A000726, A008485, A121596, A128758, A270913, A285927, A296044, A296163.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 06 2017
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