A351761 - OEIS

%I #21 Feb 19 2022 13:58:42

%S 1,1,0,1,1,0,1,2,4,0,1,3,12,21,0,1,4,24,102,148,0,1,5,40,279,1160,

%T 1305,0,1,6,60,588,4332,16490,13806,0,1,7,84,1065,11536,84075,281292,

%U 170401,0,1,8,112,1746,25220,282900,1958058,5598110,2403640,0

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

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

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

%e Square array begins:

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

%e   0,    1,     2,     3,      4,      5, ...

%e   0,    4,    12,    24,     40,     60, ...

%e   0,   21,   102,   279,    588,   1065, ...

%e   0,  148,  1160,  4332,  11536,  25220, ...

%e   0, 1305, 16490, 84075, 282900, 746525, ...

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

%o (PARI) T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

%Y Columns k=0..3 give A000007, A006153, A351762, A351763.

%Y Main diagonal gives A351765.

%Y Cf. A292860, A351776.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Feb 18 2022