A270915 -id:A270915 - cp's OEIS frontend

A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

1, 0, 3, 19, 101, 501, 2486, 12398, 62329, 315436, 1605330, 8207552, 42124368, 216903051, 1119974861, 5796944342, 30068145889, 156250892593, 813310723907, 4239676354631, 22130265931880, 115654632452514, 605081974091853, 3168828466966365, 16610409114771876, 87141919856550506

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Number of partitions of n into 2 or more parts of n kinds. - Ilya Gutkovskiy, May 16 2018

Index entries for sequences related to partitions

Cf. A000065, A008485, A022567, A093160, A270913, A285927, A285928, A286653, A296044, A296162, A296163, A304626.

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

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724... and c = 0.268015212710733315686... - Vaclav Kotesovec, May 16 2018

A366022 Decimal expansion of a constant related to the asymptotics of A109085.

Original entry on oeis.org

4, 8, 9, 6, 3, 5, 2, 2, 6, 6, 8, 4, 3, 0, 3, 3, 7, 3, 0, 8, 1, 5, 4, 1, 6, 6, 0, 5, 7, 8, 4, 6, 8, 6, 1, 9, 3, 2, 2, 4, 1, 6, 6, 2, 5, 1, 0, 1, 1, 5, 8, 7, 8, 4, 5, 4, 9, 4, 0, 6, 7, 2, 9, 9, 7, 0, 5, 7, 5, 8, 4, 1, 5, 7, 1, 4, 0, 1, 6, 8, 3, 2, 8, 8, 7, 0, 5, 2, 2, 9, 0, 1, 9, 6, 3, 9, 3, 8, 9, 9, 1, 7, 3, 2, 7, 6

Offset: 0

Vaclav Kotesovec, Sep 26 2023

			0.489635226684303373081541660578468619322416625...

Cf. A109085, A270915, A366018.

val = -s*Log[r*s] / Sqrt[2*Pi*((-2 - 3*Log[r*s] + 2*Log[1 - r*s])* QPolyGamma[0, 1, r*s] + QPolyGamma[0, 1, r*s]^2 - 4*ArcTanh[1 - 2*r*s]*(Log[r*s] - Log[1 - r*s]/2 - r*(s/(1 - r*s))) - 2*(Log[1 - r*s]/(1 - r*s)) - QPolyGamma[1, 1, r*s] + r*s*Log[r* s]*((-r)*s^2*Log[r*s]* Derivative[0, 2][QPochhammer][r*s, r*s] + 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))] /. FindRoot[{s == 1/QPochhammer[r*s], 1/s + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/(s* Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 1000]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]]

Equals limit_{n->infinity} A109085(n) * n^(3/2) / A270915^n.

cp's OEIS Frontend

A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

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