%I #10 Dec 28 2020 09:53:43

%S 42,1485,34034,647920,11187462,182587701,2880017910,44477796451,

%T 677940669900,10250875770135,154278143783022,2316262521915440,

%U 34742240691197182,521131993897607925,7822497290908844702,117554364707534272375,1769075045150700563052

%N 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 Alois P. Heinz, <a href="/A258492/b258492.txt">Table of n, a(n) for n = 5..800</a>

%F a(n) ~ 16^n / (54*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015

%p A:= proc(n, k) option remember; `if`(n=0, 1, k/n*

%p add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))

%p end:

%p T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):

%p a:= n-> T(n, 5):

%p seq(a(n), n=5..25);

%t 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 T[n_, k_] := Sum[(-1)^i A[n, k - i]/(i! (k - i)!), {i, 0, k}];

%t a[n_] := T[n, 5];

%t a /@ Range[5, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)

%Y Column k=5 of A256117.

%K nonn

%O 5,1

%A _Alois P. Heinz_, May 31 2015