A340987 - OEIS

%I #24 Jan 15 2024 12:53:41

%S 1,2,10,59,362,2287,14719,95965,631714,4189334,27946335,187319827,

%T 1260570515,8511460908,57634550179,391232510284,2661483301282,

%U 18140003082945,123846214549072,846801764644618,5797865791444367,39745254613927264,272762265331208465

%N Number of colored integer partitions of 2n such that all colors from an n-set are used.

%H Alois P. Heinz, <a href="/A340987/b340987.txt">Table of n, a(n) for n = 0..1183</a>

%F a(n) = [x^(2n)] (-1 + Product_{j>0} 1/(1-x^j))^n.

%F a(n) = A060642(2*n,n).

%F a(n) = Sum_{i=0..n} (-1)^i * C(n,i) * A144064(2n,n-i).

%F a(n) ~ c * d^n / sqrt(n), where d = 7.0224714601856191637116674203375767768930294104680988528373522936595686998... and c = 0.306577097117652483059452115503859901867921865482563952948772592499558... - _Vaclav Kotesovec_, Feb 14 2021

%e a(1) = 2: 2a, 1a1a.

%e a(2) = 10: 3a1b, 3b1a, 2a2b, 2a1b1b, 2b1a1a, 2a1a1b, 2b1a1b, 1a1b1b1b, 1a1a1b1b, 1a1a1a1b.

%p b:= proc(n, k) option remember; `if`(k<2, combinat[numbpart](n+1),

%p       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..25);

%t b[n_, k_] := b[n, k] = If[k<2, PartitionsP[n+1], With[{q = Quotient[k, 2]}, Sum[b[j, q] b[n-j, k-q], {j, 0, n}]]];

%t a[n_] := b[n, n];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Feb 04 2021, after _Alois P. Heinz_ *)

%t Table[SeriesCoefficient[(-1 + 1/QPochhammer[Sqrt[x]])^n, {x, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Jan 15 2024 *)

%t (* Calculation of constant d: *) 1/r/.FindRoot[{1 + s == 1/QPochhammer[Sqrt[r*s]], 1/(1 + s) + Sqrt[r]*(1 + s)*Derivative[0, 1][QPochhammer][Sqrt[r*s], Sqrt[r*s]] / (2*Sqrt[s]) == (Log[1 - Sqrt[r*s]] + QPolyGamma[0, 1, Sqrt[r*s]]) / (s*Log[r*s])}, {r, 1/7}, {s, 1}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Jan 15 2024 *)

%Y Cf. A000041, A060642, A144064, A324595.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Feb 01 2021