OFFSET
0,13
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
EXAMPLE
A(2,3) = 3: {aa}, {ab}, {ba}.
A(3,2) = 7: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}.
A(3,3) = 13: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 3, 3, 3, 3, 3, 3, 3, ...
0, 2, 7, 13, 13, 13, 13, 13, 13, ...
0, 2, 18, 36, 60, 60, 60, 60, 60, ...
0, 3, 42, 122, 206, 326, 326, 326, 326, ...
0, 4, 110, 433, 865, 1345, 2065, 2065, 2065, ...
0, 5, 250, 1356, 3408, 6228, 9468, 14508, 14508, ...
0, 6, 627, 4449, 15025, 29845, 51325, 76525, 116845, ...
MAPLE
b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
add(b(n-j, j, t-1)/j!, j=i..n/t))
end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
end:
A:= (n, k)-> h(n$2, min(n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
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[g[i, k], j], {j, 0, n/i}]]];
A[n_, k_] := h[n, n, Min[n, k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten(* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 23 2017
STATUS
approved