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A306025 Number of length-n restricted growth strings (RGS) with growth <= n and first element in [n].

%I A306025 #13 Sep 07 2022 04:10:36
%S A306025 1,1,7,95,2096,67354,2943277,166862583,11858631472,1029154793775,
%T A306025 106837050484924,13046411412001307,1848336205780389404,
%U A306025 300289842081446066173,55393980428260038660617,11503469972529028999979343,2669299049110696359069533376
%N A306025 Number of length-n restricted growth strings (RGS) with growth <= n and first element in [n].
%H A306025 Alois P. Heinz, <a href="/A306025/b306025.txt">Table of n, a(n) for n = 0..265</a>
%F A306025 a(n) = n! * [x^n] exp(Sum_{j=1..n} (exp(j*x)-1)/j).
%F A306025 a(n) = A306024(n,n).
%e A306025 a(0) = 1: the empty string.
%e A306025 a(1) = 1: 1.
%e A306025 a(2) = 7: 11, 12, 13, 21, 22, 23, 24.
%p A306025 b:= proc(n, k, m) option remember; `if`(n=0, 1,
%p A306025       add(b(n-1, k, max(m, j)), j=1..m+k))
%p A306025     end:
%p A306025 a:= n-> b(n$2, 0):
%p A306025 seq(a(n), n=0..20);
%p A306025 # second Maple program:
%p A306025 a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..n)), x, n+1), x, n):
%p A306025 seq(a(n), n=0..20);
%t A306025 b[n_, k_, m_] := b[n, k, m] = If[n == 0, 1,
%t A306025    Sum[b[n - 1, k, Max[m, j]], {j, 1, m + k}]];
%t A306025 a[n_] := b[n, n, 0];
%t A306025 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Sep 07 2022, after _Alois P. Heinz_ *)
%Y A306025 Main diagonal of A306024.
%Y A306025 Cf. A305963.
%K A306025 nonn
%O A306025 0,3
%A A306025 _Alois P. Heinz_, Jun 17 2018