OFFSET
0,12
FORMULA
G.f. for row n of table: Product_{j>=1} Sum_{k=0..n} x^(j*k) = Product_{j>=1} (1-x^((n+1)*j)) / (1-x^j). - Sean A. Irvine, Jan 26 2023
EXAMPLE
Square table T(n,k) begins:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ...
1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, ...
1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, ...
1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ...
1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ...
1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, ...
1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
end:
A:= (n, k)-> b(k$2, n):
seq(seq(A(n, d-n), n=0..d), d=0..13); # Alois P. Heinz, Jan 26 2023
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, k], {j, 0, Min[n/i, k]}]]];
A[n_, k_] := b[k, k, n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Feb 11 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Apr 20 2001
STATUS
approved