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%I #11 Sep 27 2023 16:20:57
%S 1,1,34,3509,657370,182587701,67773956250,31600247019120,
%T 17769492060922914,11710509049983422030,8855064908059488718600,
%U 7558849413204728468703991,7190781941414575290014093320,7544364858457252265315311530675,8654711454787575656983217747533920
%N Number of words of length 4n such that all letters of the n-ary 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="/A258499/b258499.txt">Table of n, a(n) for n = 0..250</a>
%F a(n) = A256117(2n,n).
%F a(n) ~ c * d^n * n! / n^(5/2), where d = A256254 = 98.82487375173568573170688..., c = -sqrt(2) * LambertW(-2*exp(-2)) / (16 * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.008372249434869139279228556376854454452398... . - _Vaclav Kotesovec_, Jun 01 2015, updated Sep 27 2023
%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(2*n, n):
%p seq(a(n), n=0..20);
%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[2n, n];
%t a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)
%Y Cf. A256117.
%K nonn
%O 0,3
%A _Alois P. Heinz_, May 31 2015
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