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.

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

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

Columns k=0-10 give: A000007, A000110, A002872, A306027, A306028, A306029, A306030, A306031, A306032, A306033, A306034.

Main diagonal gives A306025.

Antidiagonal sums give A306026.

Cf. A305962.

Sequence in context: A261780 A124540 A124550 * A237018 A290605 A292913

Adjacent sequences: A306021 A306022 A306023 * A306025 A306026 A306027

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 17 2018

Status

approved