OFFSET
0,9
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
EXAMPLE
A(2,3) = 4: {aa}, {ab}, {ba}, {a,a}.
A(3,2) = 8: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.
A(3,3) = 14: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}, {a,a,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, 2, 4, 4, 4, 4, 4, 4, 4, ...
0, 3, 8, 14, 14, 14, 14, 14, 14, ...
0, 5, 25, 43, 67, 67, 67, 67, 67, ...
0, 7, 53, 139, 223, 343, 343, 343, 343, ...
0, 11, 148, 495, 951, 1431, 2151, 2151, 2151, ...
0, 15, 328, 1544, 3680, 6620, 9860, 14900, 14900, ...
0, 22, 858, 5111, 16239, 31539, 53739, 78939, 119259, ...
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)):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
end:
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]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*g[d, k], {d, Divisors[j]}]* A[n - j, k], {j, 1, n}]/n];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 07 2018, from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 21 2017
STATUS
approved