OFFSET
0,6
LINKS
Alois P. Heinz, Antidiagonals n = 0..50, flattened
FORMULA
A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n.
EXAMPLE
A(3,2) = 93: (z1+z2+z3)^3 = z1^3 +3*z1^2*z2 +3*z1^2*z3 +3*z1*z2^2 +6*z1*z2*z3 +3*z1*z3^2 +z2^3 +3*z2^2*z3 +3*z2*z3^2 +z3^3 => 1^2+3^2+3^2+3^2+6^2+3^2+1^2+3^2+3^2+1^2 = 93.
Square array A(n,k) begins:
0 : 1, 1, 1, 1, 1, 1, ...
1 : 1, 1, 1, 1, 1, 1, ...
2 : 3, 4, 6, 10, 18, 34, ...
3 : 10, 27, 93, 381, 1785, 9237, ...
4 : 35, 256, 2716, 36628, 591460, 11007556, ...
5 : 126, 3125, 127905, 7120505, 495872505, 41262262505, ...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] * Binomial[n, j]^(k-1)/j!, {j, 0, n}]]]; A[n_, k_] := n!*b[n, n, k]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 21 2014
STATUS
approved