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A258499 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. 3

%I

%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 = 0.008372249434869139279228556376854454452398... . - _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(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-Fran├žois 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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Last modified April 20 20:26 EDT 2021. Contains 343137 sequences. (Running on oeis4.)