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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.

%I A261784 #22 Apr 20 2024 04:51:46
%S A261784 1,2,66,5546,893490,235804122,92540869002,50592275219138,
%T A261784 36763980389367378,34277110454602760762,39890088439337327537706,
%U A261784 56678337951284473917309346,96562013312452672907356749786,194303876852797223949281552591106,455927121076167354458618221923117282
%N 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.
%H A261784 Alois P. Heinz, <a href="/A261784/b261784.txt">Table of n, a(n) for n = 0..200</a>
%F A261784 a(n) = A261781(2n,n).
%F A261784 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
%p A261784 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A261784       add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
%p A261784     end:
%p A261784 a:= n-> add(A(2*n, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A261784 seq(a(n), n=0..15);
%t A261784 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 *)
%Y A261784 Cf. A261781, A261828.
%K A261784 nonn
%O A261784 0,2
%A A261784 _Alois P. Heinz_, Aug 31 2015