OFFSET
0,8
COMMENTS
A(n,k) counts strings [s_1, ..., s_n] with 1 <= s_i <= k + max(0, max_{j<i} s_j).
LINKS
Alois P. Heinz, Antidiagonals n = 0..150, flattened
FORMULA
E.g.f. of column k: exp(Sum_{j=1..k} (exp(j*x)-1)/j).
EXAMPLE
A(2,3) = 15: 11, 12, 13, 14, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 36.
A(4,1) = 15: 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 1234.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 2, 7, 15, 26, 40, 57, 77, ...
0, 5, 31, 95, 214, 405, 685, 1071, ...
0, 15, 164, 717, 2096, 4875, 9780, 17689, ...
0, 52, 999, 6221, 23578, 67354, 160201, 335083, ...
0, 203, 6841, 60619, 297692, 1044045, 2943277, 7117789, ...
MAPLE
b:= proc(n, k, m) option remember; `if`(n=0, 1,
add(b(n-1, k, max(m, j)), j=1..m+k))
end:
A:= (n, k)-> b(n, k, 0):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= (n, k)-> n!*coeff(series(exp(add(
(exp(j*x)-1)/j, j=1..k)), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
b[n_, k_, m_] := b[n, k, m] = If[n==0, 1, Sum[b[n-1, k, Max[m, j]], {j, 1, m+k}]];
A[n_, k_] := b[n, k, 0];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 17 2018
STATUS
approved