A305965 -id:A305965 - 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.

A306029 Number of length-n restricted growth strings (RGS) with growth <= five and first element in [5].

Original entry on oeis.org

1, 5, 40, 405, 4875, 67354, 1044045, 17867125, 333554020, 6730070329, 145676361731, 3362266525430, 82326965117385, 2129349953723509, 57961263778376192, 1655067729384150829, 49437118345913831595, 1540860755766376984434, 50000885646431513577973

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

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

Column k=5 of A306024.

Cf. A305965.

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, 0):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..5)), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=1..5} (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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