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A292503 Number of partitions of n with n sorts of part 1 which are introduced in ascending order.

%I A292503 #14 May 17 2018 13:44:00
%S A292503 1,1,3,7,20,63,233,966,4454,22404,121616,706362,4361977,28494493,
%T A292503 196087988,1416515642,10709058487,84505818259,694397612486,
%U A292503 5929368380664,52513737017847,481577858196052,4565851595293151,44692014464166068,451058715629365617
%N A292503 Number of partitions of n with n sorts of part 1 which are introduced in ascending order.
%H A292503 Alois P. Heinz, <a href="/A292503/b292503.txt">Table of n, a(n) for n = 0..250</a>
%e A292503 a(2) = 3: 2, 1a1a, 1a1b.
%e A292503 a(3) = 7: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1a1b1c.
%p A292503 f:= (n, k)-> add(Stirling2(n, j), j=0..k):
%p A292503 b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
%p A292503       f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
%p A292503     end:
%p A292503 a:= n-> b(n$3):
%p A292503 seq(a(n), n=0..30);
%t A292503 f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
%t A292503 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, f[n, k], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
%t A292503 a[n_] := b[n, n, n];
%t A292503 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%Y A292503 Cf. A292462, A292463, A292507.
%Y A292503 Main diagonal of A292745.
%Y A292503 Row sums of A292746.
%K A292503 nonn
%O A292503 0,3
%A A292503 _Alois P. Heinz_, Sep 17 2017