login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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);

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 22 04:32 EDT 2019. Contains 321406 sequences. (Running on oeis4.)