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A261783 Number of compositions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order.

%I A261783 #20 Aug 06 2024 09:56:41
%S A261783 1,1,7,73,1031,18501,403495,10366833,306717703,10271072557,
%T A261783 384058268507,15861842372465,717135437119271,35228475333207937,
%U A261783 1868440035684996207,106412817671933423073,6477200889282232394759,419626639092214594301373,28829330550533269570699411
%N A261783 Number of compositions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order.
%H A261783 Alois P. Heinz, <a href="/A261783/b261783.txt">Table of n, a(n) for n = 0..365</a>
%F A261783 a(n) = A261780(n,n).
%F A261783 a(n) = [x^n] (1-x)^n/(2*(1-x)^n-1).
%F A261783 a(n) ~ n^n / (sqrt(2) * (log(2))^(n+1)). - _Vaclav Kotesovec_, Sep 21 2019
%F A261783 a(n) = Sum_{k>=1} (1/2)^k * binomial(k*n-1,n). - _Seiichi Manyama_, Aug 06 2024
%p A261783 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A261783       add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
%p A261783     end:
%p A261783 a:= n-> A(n$2):
%p A261783 seq(a(n), n=0..20);
%t A261783 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_] := A[n, n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 24 2017, translated from Maple *)
%Y A261783 Main diagonal of A261780.
%Y A261783 Cf. A209668.
%K A261783 nonn
%O A261783 0,3
%A A261783 _Alois P. Heinz_, Aug 31 2015