A292845 Total number of words beginning with the first letter of an n-ary alphabet in all sets of nonempty words with a total of n letters.

0, 1, 3, 28, 325, 4976, 92869, 2038842, 51397801, 1461081781, 46192638386, 1606531631321, 60921659773609, 2500525907856718, 110403919405245712, 5216038547426332891, 262495788417549517393, 14015335940464667636300, 791161963786588958170705

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..382

Example

For n=2 and alphabet {a,b} we have 5 sets: {aa}, {ab}, {ba}, {bb}, {a,b}. There is a total of 3 words beginning with the first alphabet letter, thus a(2) = 3.

Maple

h:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add(
     (p-> p+[0, p[1]*j])(binomial(k^i, j)*h(n-i*j, i-1, k)), j=0..n/i)))
    end:
a= n-> `if`(n=0, 0, h(n$3)[2]/n):
seq(a(n), n=0..22);

Mathematica

h[n_, i_, k_] := h[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[ Function[p, p + {0, p[[1]]*j}][Binomial[k^i, j]*h[n - i*j, i - 1, k]], {j, 0, n/i}]]];
a[n_] := If[n == 0, 0, h[n, n, n][[2]]/n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

Crossrefs

Cf. A216158, A292804, A292805, A292873.

Sequence in context: A277507 A176335 A072343 * A212032 A352383 A151423

Adjacent sequences: A292842 A292843 A292844 * A292846 A292847 A292848

Keyword

nonn

Author

Alois P. Heinz, Sep 24 2017

Status

approved