A306025 -id:A306025 - cp's OEIS frontend

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

Alois P. Heinz, Jun 17 2018

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

			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, ...

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

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.

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);

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 *)

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

A305963 Number of length-n restricted growth strings (RGS) with growth <= n and fixed first element.

Original entry on oeis.org

1, 1, 3, 22, 305, 6756, 216552, 9416240, 530764089, 37498693555, 3235722405487, 334075729235172, 40587204883652869, 5722676826879812177, 925590727478445526747, 170032646641380554970304, 35173161711207720944899921, 8132124409499796317194563900

Offset: 0

Alois P. Heinz, Jun 15 2018

nonn

			a(2) = 3: 11, 12, 13.
a(3) = 22: 111, 112, 113, 114, 121, 122, 123, 124, 125, 131, 132, 133, 134, 135, 136, 141, 142, 143, 144, 145, 146, 147.

Alois P. Heinz, Table of n, a(n) for n = 0..266

Main diagonal of A305962.

Cf. A306025.

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-> b(n$2, 1-n):
seq(a(n), n=0..20);
# second Maple program:
a:= n-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
           (exp(j*x)-1)/j, j=1..n)), x, n), x, n-1)):
seq(a(n), n=0..20);

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_] := b[n, n, 1 - n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 21 2022, after Alois P. Heinz *)

a(n) = (n-1)! * [x^(n-1)] exp(x + Sum_{j=1..n} (exp(j*x)-1)/j) for n > 0, a(0) = 1.

a(n) = A305962(n,n).

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cp's OEIS Frontend

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.

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