OFFSET
0,8
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
G.f. of column k: Product_{j>=1} (1+x^j)^(k^j).
A(n,k) = Sum_{i=0..k} C(k,i) * A319501(n,i).
EXAMPLE
A(2,2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 1, 5, 12, 22, 35, 51, 70, ...
0, 2, 16, 55, 132, 260, 452, 721, ...
0, 2, 42, 225, 729, 1805, 3777, 7042, ...
0, 3, 116, 927, 4000, 12376, 31074, 67592, ...
0, 4, 310, 3729, 21488, 83175, 250735, 636517, ...
0, 5, 816, 14787, 113760, 550775, 1993176, 5904746, ...
MAPLE
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
end:
A:= (n, k)-> h(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
h[n_, i_, k_] := h[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[h[n-i*j, i-1, k]* Binomial[k^i, j], {j, 0, n/i}]]];
A[n_, k_] := h[n, n, k];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 23 2017
STATUS
approved