cp's OEIS Frontend

A340987 Number of colored integer partitions of 2n such that all colors from an n-set are used.

%I A340987 #24 Jan 15 2024 12:53:41
%S A340987 1,2,10,59,362,2287,14719,95965,631714,4189334,27946335,187319827,
%T A340987 1260570515,8511460908,57634550179,391232510284,2661483301282,
%U A340987 18140003082945,123846214549072,846801764644618,5797865791444367,39745254613927264,272762265331208465
%N A340987 Number of colored integer partitions of 2n such that all colors from an n-set are used.
%H A340987 Alois P. Heinz, <a href="/A340987/b340987.txt">Table of n, a(n) for n = 0..1183</a>
%F A340987 a(n) = [x^(2n)] (-1 + Product_{j>0} 1/(1-x^j))^n.
%F A340987 a(n) = A060642(2*n,n).
%F A340987 a(n) = Sum_{i=0..n} (-1)^i * C(n,i) * A144064(2n,n-i).
%F A340987 a(n) ~ c * d^n / sqrt(n), where d = 7.0224714601856191637116674203375767768930294104680988528373522936595686998... and c = 0.306577097117652483059452115503859901867921865482563952948772592499558... - _Vaclav Kotesovec_, Feb 14 2021
%e A340987 a(1) = 2: 2a, 1a1a.
%e A340987 a(2) = 10: 3a1b, 3b1a, 2a2b, 2a1b1b, 2b1a1a, 2a1a1b, 2b1a1b, 1a1b1b1b, 1a1a1b1b, 1a1a1a1b.
%p A340987 b:= proc(n, k) option remember; `if`(k<2, combinat[numbpart](n+1),
%p A340987       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
%p A340987     end:
%p A340987 a:= n-> b(n$2):
%p A340987 seq(a(n), n=0..25);
%t A340987 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 A340987 a[n_] := b[n, n];
%t A340987 a /@ Range[0, 25] (* _Jean-François Alcover_, Feb 04 2021, after _Alois P. Heinz_ *)
%t A340987 Table[SeriesCoefficient[(-1 + 1/QPochhammer[Sqrt[x]])^n, {x, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Jan 15 2024 *)
%t A340987 (* 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 A340987 Cf. A000041, A060642, A144064, A324595.
%K A340987 nonn
%O A340987 0,2
%A A340987 _Alois P. Heinz_, Feb 01 2021