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A261828 Number of compositions of 2n into distinct parts 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 A261828 #17 Feb 25 2017 03:00:44
%S A261828 1,1,15,832,14791,2008546,55380132,2868333476,511805155863,
%T A261828 31512728488918,2638310862477610,926651539894899446,
%U A261828 74254761492776175196,6851495812540548188072,9541620342114654822145972,611287722968440282212322702,58354641005988089624088037623
%N A261828 Number of compositions of 2n into distinct parts 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 A261828 Alois P. Heinz, <a href="/A261828/b261828.txt">Table of n, a(n) for n = 0..150</a>
%F A261828 a(n) = A261836(2n,n).
%p A261828 b:= proc(n, i, p, k) option remember;
%p A261828       `if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p, k)+
%p A261828       `if`(i>n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
%p A261828     end:
%p A261828 a:= n-> add(b(2*n$2, 0, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A261828 seq(a(n), n=0..20);
%t A261828 b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2<n, 0, If[n==0, p!, b[n, i - 1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; a[n_] := Sum[b[2*n, 2*n, 0, n-i]*(-1)^i*Binomial[n, i], {i, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 25 2017, translated from Maple *)
%Y A261828 Cf. A261784, A261836.
%K A261828 nonn
%O A261828 0,3
%A A261828 _Alois P. Heinz_, Sep 02 2015