OFFSET
0,8
LINKS
Alois P. Heinz, Antidiagonals n = 0..200, flattened
Jessica Jay and Benjamin Lees, Combinatorial identities from an inhomogeneous Ising chain, arXiv:2401.16311 [math.PR], 2024.
Wikipedia, Partition (number theory)
FORMULA
G.f. of column k: Product_{j>=1} (1+(k-1)*x^j)/(1-x^j).
A(n,k) = Sum_{i=0..floor((sqrt(1+8*k)-1)/2)} k!/(k-i)! * A321878(n,i).
EXAMPLE
A(3,2) = 8: 3a, 3b, 2a1a, 2a1b, 2b1a, 2b1b, 111a, 111b.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, ...
0, 2, 4, 6, 8, 10, 12, 14, 16, ...
0, 3, 8, 15, 24, 35, 48, 63, 80, ...
0, 5, 14, 27, 44, 65, 90, 119, 152, ...
0, 7, 24, 51, 88, 135, 192, 259, 336, ...
0, 11, 40, 93, 176, 295, 456, 665, 928, ...
0, 15, 64, 159, 312, 535, 840, 1239, 1744, ...
0, 22, 100, 264, 544, 970, 1572, 2380, 3424, ...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 27 2019
STATUS
approved