A305963 Number of length-n restricted growth strings (RGS) with growth <= n and fixed first element.

1, 1, 3, 22, 305, 6756, 216552, 9416240, 530764089, 37498693555, 3235722405487, 334075729235172, 40587204883652869, 5722676826879812177, 925590727478445526747, 170032646641380554970304, 35173161711207720944899921, 8132124409499796317194563900

Offset: 0

Author

Alois P. Heinz, Jun 15 2018

Keywords

• nonn

Examples

			a(2) = 3: 11, 12, 13.
a(3) = 22: 111, 112, 113, 114, 121, 122, 123, 124, 125, 131, 132, 133, 134, 135, 136, 141, 142, 143, 144, 145, 146, 147.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..266

Crossrefs

Main diagonal of A305962.

Cf. A306025.

Programs

Maple

b:= proc(n, k, m) option remember; `if`(n=0, 1,
      add(b(n-1, k, max(m, j)), j=1..m+k))
    end:
a:= n-> b(n$2, 1-n):
seq(a(n), n=0..20);
# second Maple program:
a:= n-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
           (exp(j*x)-1)/j, j=1..n)), x, n), x, n-1)):
seq(a(n), n=0..20);

Mathematica

b[n_, k_, m_] := b[n, k, m] = If[n == 0, 1,
     Sum[b[n - 1, k, Max[m, j]], {j, 1, m + k}]];
a[n_] := b[n, n, 1 - n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 21 2022, after Alois P. Heinz *)

Formula

a(n) = (n-1)! * [x^(n-1)] exp(x + Sum_{j=1..n} (exp(j*x)-1)/j) for n > 0, a(0) = 1.

a(n) = A305962(n,n).