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.

1, 1, 34, 3509, 657370, 182587701, 67773956250, 31600247019120, 17769492060922914, 11710509049983422030, 8855064908059488718600, 7558849413204728468703991, 7190781941414575290014093320, 7544364858457252265315311530675, 8654711454787575656983217747533920

Offset: 0

Author

Alois P. Heinz, May 31 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A256117.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
      add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
    end:
T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
a:= n-> T(2*n, n):
seq(a(n), n=0..20);

Mathematica

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[n_, k_] := Sum[(-1)^i A[n, k-i]/(i! (k-i)!), {i, 0, k}];
a[n_] := T[2n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

a(n) = A256117(2n,n).

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