A305963 -id:A305963 - 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.

A306025 Number of length-n restricted growth strings (RGS) with growth <= n and first element in [n].

Original entry on oeis.org

1, 1, 7, 95, 2096, 67354, 2943277, 166862583, 11858631472, 1029154793775, 106837050484924, 13046411412001307, 1848336205780389404, 300289842081446066173, 55393980428260038660617, 11503469972529028999979343, 2669299049110696359069533376

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

			a(0) = 1: the empty string.
a(1) = 1: 1.
a(2) = 7: 11, 12, 13, 21, 22, 23, 24.

Alois P. Heinz, Table of n, a(n) for n = 0..265

Main diagonal of A306024.

Cf. A305963.

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, 0):
seq(a(n), n=0..20);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..n)), x, n+1), x, n):
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, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 07 2022, after Alois P. Heinz *)

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

a(n) = A306024(n,n).

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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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