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!.
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, 1, 1, 7, 51, 373, 2293, 9051, 13327, 1, 1, 8, 67, 586, 4721, 29176, 106843, 130922, 1, 1, 9, 85, 859, 8481, 70981, 427189, 1425495, 1441729, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, 6, ... 1, 7, 15, 25, 37, 51, ... 1, 34, 103, 214, 373, 586, ... 1, 209, 885, 2293, 4721, 8481, ... 1, 1546 ,9051, 29176, 70981, 146046, ...
Crossrefs
Programs
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PARI
T(n,k) = n! * sum(j=0, n, binomial(n+(k-1)*(j+1), n-j)/j!);
Formula
E.g.f. of column k: exp( x/(1-x)^k ) / (1-x)^k.
T(n,k) = Sum_{j=0..n} (n+(k-1)*(j+1))!/(k*j+k-1)! * binomial(n,j) for k > 0.