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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. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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

LINKS

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

FORMULA

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.

EXAMPLE

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, ...

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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.

Sequence in context: A124530 A243631 A070914 * A144150 A124560 A290759

Adjacent sequences: A305959 A305960 A305961 * A305963 A305964 A305965

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 15 2018

STATUS

approved

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Last modified January 29 23:01 EST 2023. Contains 359939 sequences. (Running on oeis4.)