OFFSET
0,5
COMMENTS
T(0,0) = 1 by convention.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
E.g.f. of column k>0: exp((-LambertW(-x))^k/k), e.g.f. of column k=0: 1.
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 16, 6, 2;
0, 125, 51, 24, 6;
0, 1296, 560, 300, 120, 24;
0, 16807, 7575, 4360, 2160, 720, 120;
0, 262144, 122052, 73710, 41160, 17640, 5040, 720;
...
MAPLE
with(combinat):
T:= (n, k)-> `if`(k*n=0, `if`(k+n=0, 1, 0),
add(binomial(n-1, j*k-1)*n^(n-j*k)*(k-1)!^j*
multinomial(j*k, k$j, 0)/j!, j=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
multinomial[n_, k_] := n!/Times @@ (k!); T[n_, k_] := If[k*n==0, If[k+n == 0, 1, 0], Sum[Binomial[n-1, j*k-1]*n^(n-j*k)*(k-1)!^j*multinomial[j*k, Append[Array[k&, j], 0]]/j!, {j, 0, n/k}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 18 2014
STATUS
approved