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A292745 Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A292745 #22 Oct 20 2018 14:16:22
%S A292745 1,1,0,1,1,1,1,1,2,1,1,1,3,3,2,1,1,3,6,5,2,1,1,3,7,13,7,4,1,1,3,7,19,
%T A292745 26,11,4,1,1,3,7,20,52,54,15,7,1,1,3,7,20,62,151,108,22,8,1,1,3,7,20,
%U A292745 63,217,442,219,30,12,1,1,3,7,20,63,232,803,1314,439,42,14
%N A292745 Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A292745 Alois P. Heinz, <a href="/A292745/b292745.txt">Antidiagonals n = 0..140, flattened</a>
%F A292745 A(n,k) = Sum_{j=0..k} A292746(n,j).
%F A292745 A(n,k) = A(n,n) for all k >= n.
%e A292745 A(3,2) = 6: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b.
%e A292745 Square array A(n,k) begins:
%e A292745   1,  1,   1,    1,    1,    1,    1,    1,    1, ...
%e A292745   0,  1,   1,    1,    1,    1,    1,    1,    1, ...
%e A292745   1,  2,   3,    3,    3,    3,    3,    3,    3, ...
%e A292745   1,  3,   6,    7,    7,    7,    7,    7,    7, ...
%e A292745   2,  5,  13,   19,   20,   20,   20,   20,   20, ...
%e A292745   2,  7,  26,   52,   62,   63,   63,   63,   63, ...
%e A292745   4, 11,  54,  151,  217,  232,  233,  233,  233, ...
%e A292745   4, 15, 108,  442,  803,  944,  965,  966,  966, ...
%e A292745   7, 22, 219, 1314, 3092, 4158, 4425, 4453, 4454, ...
%p A292745 f:= (n, k)-> add(Stirling2(n, j), j=0..k):
%p A292745 b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
%p A292745       f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
%p A292745     end:
%p A292745 A:= (n, k)-> b(n$2, k):
%p A292745 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A292745 f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
%t A292745 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 A292745 A[n_, k_] := b[n, n, k];
%t A292745 Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%Y A292745 Columns k=0-10 give: A002865, A000041, A320733, A320734, A320735, A320736, A320737, A320738, A320739, A320740, A320741.
%Y A292745 Main diagonal gives A292503.
%Y A292745 Cf. A292508, A292622, A292741, A292746.
%K A292745 nonn,tabl
%O A292745 0,9
%A A292745 _Alois P. Heinz_, Sep 22 2017