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A258492 Number of words of length 2n such that all letters of the quinary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word.

%I A258492 #10 Dec 28 2020 09:53:43
%S A258492 42,1485,34034,647920,11187462,182587701,2880017910,44477796451,
%T A258492 677940669900,10250875770135,154278143783022,2316262521915440,
%U A258492 34742240691197182,521131993897607925,7822497290908844702,117554364707534272375,1769075045150700563052
%N A258492 Number of words of length 2n such that all letters of the quinary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word.
%H A258492 Alois P. Heinz, <a href="/A258492/b258492.txt">Table of n, a(n) for n = 5..800</a>
%F A258492 a(n) ~ 16^n / (54*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%p A258492 A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
%p A258492       add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
%p A258492     end:
%p A258492 T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
%p A258492 a:= n-> T(n, 5):
%p A258492 seq(a(n), n=5..25);
%t A258492 A[n_, k_] := A[n, k] = If[n == 0, 1, (k/n) Sum[Binomial[2n, j] (n - j)*If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];
%t A258492 T[n_, k_] := Sum[(-1)^i A[n, k - i]/(i! (k - i)!), {i, 0, k}];
%t A258492 a[n_] := T[n, 5];
%t A258492 a /@ Range[5, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A258492 Column k=5 of A256117.
%K A258492 nonn
%O A258492 5,1
%A A258492 _Alois P. Heinz_, May 31 2015