A261784 Number of compositions of 2n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur at least once in the composition.

1, 2, 66, 5546, 893490, 235804122, 92540869002, 50592275219138, 36763980389367378, 34277110454602760762, 39890088439337327537706, 56678337951284473917309346, 96562013312452672907356749786, 194303876852797223949281552591106, 455927121076167354458618221923117282

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) = A261781(2n,n).

a(n) ~ c * d^n * n!^2 / n, where d = -4 / (log(2)^2 * LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 12.85568917366612131932660300054233483234... and c = 0.25886492311146555025177523170232718427705044811049100445591... . - Vaclav Kotesovec, Feb 18 2017, updated Apr 20 2024

Maple

A:= proc(n, k) option remember; `if`(n=0, 1,
      add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> add(A(2*n, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..15);

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n - j, k]*Binomial[j + k - 1, k - 1], {j, 1, n}]]; a[n_] := Sum[A[2*n, n - i]*(-1)^i*Binomial[n, i], {i, 0, n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 25 2017, translated from Maple *)

Crossrefs

Cf. A261781, A261828.

Sequence in context: A092884 A230735 A361884 * A350730 A255435 A335546

Adjacent sequences: A261781 A261782 A261783 * A261785 A261786 A261787

Keyword

nonn

Author

Alois P. Heinz, Aug 31 2015

Status

approved