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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!.
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%I #14 Mar 07 2023 10:37:15

%S 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 61,241,581,331,1,1,1,13,91,481,1981,3661,1303,1,1,1,15,127,841,4881,

%U 17551,26335,5937,1,1,1,17,169,1345,10001,55321,171697,202049,26785,1

%N 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 E.g.f. of column k: exp(x * (1+x)^k).

%F 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 Square array begins:

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

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

%e 1, 3, 5, 7, 9, 11, ...

%e 1, 7, 19, 37, 61, 91, ...

%e 1, 25, 97, 241, 481, 841, ...

%e 1, 81, 581, 1981, 4881, 10001, ...

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

%Y Columns k=0..4 give A000012, A047974, A361278, A361279, A361280.

%Y Main diagonal gives A361281.

%Y Cf. A293012.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, Mar 06 2023