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A258491 Number of words of length 2n such that all letters of the quaternary 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 A258491 #10 Dec 28 2020 09:53:37
%S A258491 14,300,4400,55692,657370,7488228,83752760,928406556,10254052556,
%T A258491 113186465340,1250820198264,13852280754980,153813849202674,
%U A258491 1712835575525140,19129590267619304,214261857777632700,2406509409480345364,27100348605141932540,305944173898725745944
%N A258491 Number of words of length 2n such that all letters of the quaternary 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 A258491 Alois P. Heinz, <a href="/A258491/b258491.txt">Table of n, a(n) for n = 4..900</a>
%F A258491 a(n) ~ 12^n / (8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%p A258491 A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
%p A258491       add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
%p A258491     end:
%p A258491 T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
%p A258491 a:= n-> T(n, 4):
%p A258491 seq(a(n), n=4..25);
%t A258491 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 A258491 T[n_, k_] := Sum[(-1)^i A[n, k - i]/(i! (k - i)!), {i, 0, k}];
%t A258491 a[n_] := T[n, 4];
%t A258491 a /@ Range[4, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A258491 Column k=4 of A256117.
%K A258491 nonn
%O A258491 4,1
%A A258491 _Alois P. Heinz_, May 31 2015