OFFSET
0,9
COMMENTS
A(n,k) counts strings [s_1, ..., s_n] with 1 = s_1 <= s_i <= k + max_{j<i} s_j.
LINKS
Alois P. Heinz, Antidiagonals n = 0..150, flattened
FORMULA
A(n,k) = (n-1)! * [x^(n-1)] exp(x+Sum_{j=1..k} (exp(j*x)-1)/j) for n>0, A(0,k) = 1.
EXAMPLE
A(0,2) = 1: the empty string.
A(1,2) = 1: 1.
A(2,2) = 3: 11, 12, 13.
A(3,2) = 12: 111, 112, 113, 121, 122, 123, 124, 131, 132, 133, 134, 135.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, 8, ...
1, 5, 12, 22, 35, 51, 70, 92, ...
1, 15, 59, 150, 305, 541, 875, 1324, ...
1, 52, 339, 1200, 3125, 6756, 12887, 22464, ...
1, 203, 2210, 10922, 36479, 96205, 216552, 435044, ...
1, 877, 16033, 110844, 475295, 1530025, 4065775, 9416240, ...
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, 1-k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= (n, k)-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
(exp(j*x)-1)/j, j=1..k)), x, n), x, n-1)):
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, 1-k];
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 15 2018
STATUS
approved