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

0,3

Links

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

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

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 *)

Crossrefs

Cf. A256117.

Sequence in context: A252709 A138590 A069223 * A218718 A129056 A212034

Adjacent sequences: A258496 A258497 A258498 * A258500 A258501 A258502

Keyword

nonn

Author

Alois P. Heinz, May 31 2015

Status

approved