cp's OEIS Frontend

A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

%I A361277 #14 Mar 07 2023 10:37:15
%S A361277 1,1,1,1,1,1,1,1,3,1,1,1,5,7,1,1,1,7,19,25,1,1,1,9,37,97,81,1,1,1,11,
%T A361277 61,241,581,331,1,1,1,13,91,481,1981,3661,1303,1,1,1,15,127,841,4881,
%U A361277 17551,26335,5937,1,1,1,17,169,1345,10001,55321,171697,202049,26785,1
%N A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.
%F A361277 E.g.f. of column k: exp(x * (1+x)^k).
%F A361277 T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=1..n} j * binomial(k,j-1) * T(n-j,k)/(n-j)!.
%e A361277 Square array begins:
%e A361277   1,  1,   1,    1,    1,     1, ...
%e A361277   1,  1,   1,    1,    1,     1, ...
%e A361277   1,  3,   5,    7,    9,    11, ...
%e A361277   1,  7,  19,   37,   61,    91, ...
%e A361277   1, 25,  97,  241,  481,   841, ...
%e A361277   1, 81, 581, 1981, 4881, 10001, ...
%o A361277 (PARI) T(n, k) = n!*sum(j=0, n, binomial(k*j, n-j)/j!);
%Y A361277 Columns k=0..4 give A000012, A047974, A361278, A361279, A361280.
%Y A361277 Main diagonal gives A361281.
%Y A361277 Cf. A293012.
%K A361277 nonn,tabl
%O A361277 0,9
%A A361277 _Seiichi Manyama_, Mar 06 2023