A292713 Number of multisets of nonempty words with a total of n letters over n-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

1, 1, 4, 14, 67, 343, 2151, 14900, 119259, 1055520, 10465854, 113479756, 1350508150, 17373376892, 241576630993, 3596468789967, 57232276979726, 967517444008250, 17339617861447844, 328037083000497867, 6537494747743375847, 136820214583596515519

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Formula

a(n) = [x^n] Product_{j=1..n} 1/(1-x^j)^A226873(j,n).

a(n) = A292712(n,n).

a(n) ~ c * n!, where c = A247551 = 2.5294774720791526... - Vaclav Kotesovec, Oct 05 2017

Example

a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 4: {aa}, {ab}, {ba}, {a,a}.
a(3) = 14: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.

Maple

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..25);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*g[d, k], {d, Divisors[j]}]* A[n - j, k], {j, 1, n}]/n];
a[n_] := A[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

Crossrefs

Main diagonal of A292712.

Row sums of A319495.

Cf. A226873, A292796.

Sequence in context: A292723 A292724 A292725 * A007025 A221538 A301511

Adjacent sequences: A292710 A292711 A292712 * A292714 A292715 A292716

Keyword

nonn

Author

Alois P. Heinz, Sep 21 2017

Status

approved