A306029 -id:A306029 - 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).

A305965 Number of length-n restricted growth strings (RGS) with growth <= five and fixed first element.

Original entry on oeis.org

1, 1, 6, 51, 541, 6756, 96205, 1530025, 26775550, 509861195, 10472109149, 230368347780, 5396308081285, 133949699318945, 3508794554854054, 96648143868171171, 2790590111082279405, 84231759174460743700, 2651416546964399982909, 86848041397350751409257

Offset: 0

Alois P. Heinz, Jun 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..444

Column k=5 of A305962.

Cf. A306029.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+5))
    end:
a:= n-> b(n, -4):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
           (exp(j*x)-1)/j, j=1..5)), x, n), x, n-1)):
seq(a(n), n=0..25);

a(n) = (n-1)! * [x^(n-1)] exp(x+Sum_{j=1..5} (exp(j*x)-1)/j) for n>0, a(0) = 1.

Showing 1-2 of 2 results.

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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