A362839 - OEIS

%I #17 May 05 2023 12:23:18

%S 1,1,0,1,0,0,1,0,2,0,1,0,4,3,0,1,0,6,12,16,0,1,0,8,27,80,65,0,1,0,10,

%T 48,216,560,336,0,1,0,12,75,448,2025,4512,1897,0,1,0,14,108,800,5120,

%U 21708,40768,11824,0,1,0,16,147,1296,10625,67584,260253,407808,80145,0

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling2(n-j,j)/(n-j)!.

%F E.g.f. of column k: exp(x * (exp(k * x) - 1)).

%F G.f. of column k: Sum_{j>=0} x^j / (1 - (k*j-1)*x)^(j+1).

%F T(n,k) = Sum_{j=0..n} (k*j-1)^(n-j) * binomial(n,j).

%e Square array begins:

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

%e   0,  0,   0,    0,    0,     0, ...

%e   0,  2,   4,    6,    8,    10, ...

%e   0,  3,  12,   27,   48,    75, ...

%e   0, 16,  80,  216,  448,   800, ...

%e   0, 65, 560, 2025, 5120, 10625, ...

%o (PARI) T(n, k) = n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 2)/(n-j)!);

%Y Columns k=0..3 give: A000007, A052506, A351736, A351737.

%Y Main diagonal gives A356806.

%Y Cf. A362652.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, May 05 2023