A305971 -id:A305971 - 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.

A306026 Antidiagonal sums of A306024.

Original entry on oeis.org

1, 1, 2, 5, 16, 66, 343, 2180, 16505, 145773, 1477880, 16986349, 219158316, 3147962668, 49982588535, 871766923048, 16609804758449, 344016348602845, 7711752589539436, 186379711851775401, 4839449174872615116, 134575228738532130948, 3996183953610068510929

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A306024, A305971.

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

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

a(n) = Sum_{j=0..n} A306024(j,n-j).

Showing 1-2 of 2 results.

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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