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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).
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%I #39 Feb 13 2021 04:58:15

%S 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 13,73,261,331,1,1,1,3,13,73,381,1531,1303,1,1,1,3,13,73,501,2611,

%U 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 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 Seiichi Manyama, <a href="/A293669/b293669.txt">Antidiagonals n = 0..139, flattened</a>

%F A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.

%e Square array begins:

%e 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, ...

%e 1, 3, 3, 3, 3, ...

%e 1, 7, 13, 13, 13, ...

%e 1, 25, 49, 73, 73, ...

%e 1, 81, 261, 381, 501, ...

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(

%p A(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n, k)))

%p end:

%p seq(seq(A(n, 1+d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Nov 11 2020

%t 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 Table[A[n, d-n+1], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 13 2021 *)

%Y Columns k=1..9 give A000012, A047974, A118589, A193930, A193931, A193932, A193933, A306623, A306624.

%Y Rows n=0-1 give A000012.

%Y Main diagonal gives A000262.

%Y Cf. A293718, A293724.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, Oct 14 2017