A296162 a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.

1, 1, 5, 19, 89, 406, 1913, 9073, 43505, 209971, 1019390, 4971781, 24343037, 119579006, 589050663, 2908727839, 14393759457, 71360342129, 354372852011, 1762416036422, 8776797353574, 43761058841638, 218431753457637, 1091385769314272, 5458068218285909, 27319061357620056

Offset: 0

Author

Ilya Gutkovskiy, Dec 06 2017

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Partition Function b_k

Crossrefs

Cf. A000726, A008485, A121596, A128758, A270913, A285927, A296044, A296163.

Programs

Mathematica

Table[SeriesCoefficient[Product[((1 - x^(3 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
(* 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 *)

Formula

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

a(n) ~ c * d^n / sqrt(n), where d = 5.1069752682291604222843644516987970799026747758649349... and c = 0.271879273312907861082536692355942116774864... - Vaclav Kotesovec, May 13 2018