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URL: https://oeis.org/A306024

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A306024
Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and first element in [k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
15
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 31, 15, 0, 1, 5, 26, 95, 164, 52, 0, 1, 6, 40, 214, 717, 999, 203, 0, 1, 7, 57, 405, 2096, 6221, 6841, 877, 0, 1, 8, 77, 685, 4875, 23578, 60619, 51790, 4140, 0, 1, 9, 100, 1071, 9780, 67354, 297692, 652595, 428131, 21147, 0
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
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);
# Alternative:
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
Main diagonal gives A306025.
Antidiagonal sums give A306026.
Cf. A305962.
Sequence in context: A124540 A386558 A124550 * A237018 A290605 A292913
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 17 2018
STATUS
approved