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A293669 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} x^j).

%I A293669 #39 Feb 13 2021 04:58:15
%S A293669 1,1,1,1,1,1,1,1,3,1,1,1,3,7,1,1,1,3,13,25,1,1,1,3,13,49,81,1,1,1,3,
%T A293669 13,73,261,331,1,1,1,3,13,73,381,1531,1303,1,1,1,3,13,73,501,2611,
%U A293669 9073,5937,1,1,1,3,13,73,501,3331,19993,63393,26785,1,1,1,3,13,73,501,4051,27553,165873,465769,133651,1
%N A293669 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} x^j).
%H A293669 Seiichi Manyama, <a href="/A293669/b293669.txt">Antidiagonals n = 0..139, flattened</a>
%F A293669 A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.
%e A293669 Square array begins:
%e A293669    1,  1,   1,   1,   1, ...
%e A293669    1,  1,   1,   1,   1, ...
%e A293669    1,  3,   3,   3,   3, ...
%e A293669    1,  7,  13,  13,  13, ...
%e A293669    1, 25,  49,  73,  73, ...
%e A293669    1, 81, 261, 381, 501, ...
%p A293669 A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A293669       A(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n, k)))
%p A293669     end:
%p A293669 seq(seq(A(n, 1+d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Nov 11 2020
%t A293669 A[0, _] = 1; A[n_ /; n >= 0, k_ /; k >= 1] := A[n, k] = (n-1)!*Sum[j*A[n-j, k]/(n-j)!, {j, 1, Min[k, n]}]; A[_, _] = 0;
%t A293669 Table[A[n, d-n+1], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 13 2021 *)
%Y A293669 Columns k=1..9 give A000012, A047974, A118589, A193930, A193931, A193932, A193933, A306623, A306624.
%Y A293669 Rows n=0-1 give A000012.
%Y A293669 Main diagonal gives A000262.
%Y A293669 Cf. A293718, A293724.
%K A293669 nonn,tabl
%O A293669 0,9
%A A293669 _Seiichi Manyama_, Oct 14 2017