A305966 -id:A305966 - cp's OEIS frontend

A305962 Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and fixed first element; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 15, 1, 1, 1, 5, 22, 59, 52, 1, 1, 1, 6, 35, 150, 339, 203, 1, 1, 1, 7, 51, 305, 1200, 2210, 877, 1, 1, 1, 8, 70, 541, 3125, 10922, 16033, 4140, 1, 1, 1, 9, 92, 875, 6756, 36479, 110844, 127643, 21147, 1

Offset: 0

Alois P. Heinz, Jun 15 2018

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

			A(0,2) = 1: the empty string.
A(1,2) = 1: 1.
A(2,2) = 3: 11, 12, 13.
A(3,2) = 12: 111, 112, 113, 121, 122, 123, 124, 131, 132, 133, 134, 135.
Square array A(n,k) begins:
  1,   1,     1,      1,      1,       1,       1,       1, ...
  1,   1,     1,      1,      1,       1,       1,       1, ...
  1,   2,     3,      4,      5,       6,       7,       8, ...
  1,   5,    12,     22,     35,      51,      70,      92, ...
  1,  15,    59,    150,    305,     541,     875,    1324, ...
  1,  52,   339,   1200,   3125,    6756,   12887,   22464, ...
  1, 203,  2210,  10922,  36479,   96205,  216552,  435044, ...
  1, 877, 16033, 110844, 475295, 1530025, 4065775, 9416240, ...

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

Columns k=0-10 give: A000012, A000110, A080337, A189845, A305964, A305965, A305966, A305967, A305968, A305969, A305970.
Main diagonal gives: A305963.
Antidiagonal sums give: A305971.
Cf. A306024.

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, 1-k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= (n, k)-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
              (exp(j*x)-1)/j, j=1..k)), x, n), x, n-1)):
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, 1-k];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

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

A306030 Number of length-n restricted growth strings (RGS) with growth <= six and first element in [6].

Original entry on oeis.org

1, 6, 57, 685, 9780, 160201, 2943277, 59687920, 1320233315, 31557691541, 809161436022, 22121068343155, 641530646758325, 19651776950222806, 633510644286624717, 21422880077590022265, 757789084383273607060, 27969244566731240796621, 1074750913823536151018737

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..431

Column k=6 of A306024.

Cf. A305966.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+6))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..6)), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=1..6} (exp(j*x)-1)/j).

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cp's OEIS Frontend

A305962 Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and fixed first element; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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