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

0,2

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A000041, A060642, A144064, A324595.

Sequence in context: A370247 A370273 A309955 * A186758 A372415 A372476

Adjacent sequences: A340984 A340985 A340986 * A340988 A340989 A340990

Keyword

nonn

Author

Alois P. Heinz, Feb 01 2021

Status

approved