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

Author

Alois P. Heinz, Jun 17 2018

Keywords

• nonn
• tabl

Comments

A(n,k) counts strings [s_1, ..., s_n] with 1 <= s_i <= k + max(0, max_{j

Examples

			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, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..150, flattened

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.

Programs

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 *)

Formula

E.g.f. of column k: exp(Sum_{j=1..k} (exp(j*x)-1)/j).

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