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

1, 2, 10, 59, 362, 2287, 14719, 95965, 631714, 4189334, 27946335, 187319827, 1260570515, 8511460908, 57634550179, 391232510284, 2661483301282, 18140003082945, 123846214549072, 846801764644618, 5797865791444367, 39745254613927264, 272762265331208465

Offset: 0

Author

Alois P. Heinz, Feb 01 2021

Keywords

• nonn

Examples

			a(1) = 2: 2a, 1a1a.
a(2) = 10: 3a1b, 3b1a, 2a2b, 2a1b1b, 2b1a1a, 2a1a1b, 2b1a1b, 1a1b1b1b, 1a1a1b1b, 1a1a1a1b.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1183

Crossrefs

Cf. A000041, A060642, A144064, A324595.

Programs

Maple

b:= proc(n, k) option remember; `if`(k<2, combinat[numbpart](n+1),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

Mathematica

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}]]];
a[n_] := b[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Feb 04 2021, after Alois P. Heinz *)
Table[SeriesCoefficient[(-1 + 1/QPochhammer[Sqrt[x]])^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jan 15 2024 *)
(* 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 *)

Formula

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

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

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

a(n) ~ c * d^n / sqrt(n), where d = 7.0224714601856191637116674203375767768930294104680988528373522936595686998... and c = 0.306577097117652483059452115503859901867921865482563952948772592499558... - Vaclav Kotesovec, Feb 14 2021