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

%I A305963 #16 Mar 21 2022 09:34:40
%S A305963 1,1,3,22,305,6756,216552,9416240,530764089,37498693555,3235722405487,
%T A305963 334075729235172,40587204883652869,5722676826879812177,
%U A305963 925590727478445526747,170032646641380554970304,35173161711207720944899921,8132124409499796317194563900
%N A305963 Number of length-n restricted growth strings (RGS) with growth <= n and fixed first element.
%H A305963 Alois P. Heinz, <a href="/A305963/b305963.txt">Table of n, a(n) for n = 0..266</a>
%F A305963 a(n) = (n-1)! * [x^(n-1)] exp(x + Sum_{j=1..n} (exp(j*x)-1)/j) for n > 0, a(0) = 1.
%F A305963 a(n) = A305962(n,n).
%e A305963 a(2) = 3: 11, 12, 13.
%e A305963 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.
%p A305963 b:= proc(n, k, m) option remember; `if`(n=0, 1,
%p A305963       add(b(n-1, k, max(m, j)), j=1..m+k))
%p A305963     end:
%p A305963 a:= n-> b(n$2, 1-n):
%p A305963 seq(a(n), n=0..20);
%p A305963 # second Maple program:
%p A305963 a:= n-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
%p A305963            (exp(j*x)-1)/j, j=1..n)), x, n), x, n-1)):
%p A305963 seq(a(n), n=0..20);
%t A305963 b[n_, k_, m_] := b[n, k, m] = If[n == 0, 1,
%t A305963      Sum[b[n - 1, k, Max[m, j]], {j, 1, m + k}]];
%t A305963 a[n_] := b[n, n, 1 - n];
%t A305963 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 21 2022, after _Alois P. Heinz_ *)
%Y A305963 Main diagonal of A305962.
%Y A305963 Cf. A306025.
%K A305963 nonn
%O A305963 0,3
%A A305963 _Alois P. Heinz_, Jun 15 2018