cp's OEIS Frontend

A361616 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.

%I A361616 #23 Mar 26 2023 11:09:39
%S A361616 1,1,1,1,2,1,1,3,7,1,1,4,15,34,1,1,5,25,103,209,1,1,6,37,214,885,1546,
%T A361616 1,1,7,51,373,2293,9051,13327,1,1,8,67,586,4721,29176,106843,130922,1,
%U A361616 1,9,85,859,8481,70981,427189,1425495,1441729,1
%N A361616 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.
%F A361616 E.g.f. of column k: exp( x/(1-x)^k ) / (1-x)^k.
%F A361616 T(n,k) = Sum_{j=0..n} (n+(k-1)*(j+1))!/(k*j+k-1)! * binomial(n,j) for k > 0.
%e A361616 Square array begins:
%e A361616   1,    1,    1,     1,     1,      1, ...
%e A361616   1,    2,    3,     4,     5,      6, ...
%e A361616   1,    7,   15,    25,    37,     51, ...
%e A361616   1,   34,  103,   214,   373,    586, ...
%e A361616   1,  209,  885,  2293,  4721,   8481, ...
%e A361616   1, 1546 ,9051, 29176, 70981, 146046, ...
%o A361616 (PARI) T(n,k) = n! * sum(j=0, n, binomial(n+(k-1)*(j+1), n-j)/j!);
%Y A361616 Columns k=0..3 give A000012, A002720, A343884, A351767.
%Y A361616 Main diagonal gives A361617.
%Y A361616 Cf. A293985, A361600.
%K A361616 nonn,tabl
%O A361616 0,5
%A A361616 _Seiichi Manyama_, Mar 18 2023