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A243081 Number A(n,k) of compositions of n into parts with multiplicity not larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 21
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 3, 0, 1, 1, 2, 3, 3, 0, 1, 1, 2, 4, 7, 5, 0, 1, 1, 2, 4, 7, 11, 11, 0, 1, 1, 2, 4, 8, 15, 21, 13, 0, 1, 1, 2, 4, 8, 15, 26, 34, 19, 0, 1, 1, 2, 4, 8, 16, 31, 52, 59, 27, 0, 1, 1, 2, 4, 8, 16, 31, 57, 93, 114, 57, 0, 1, 1, 2, 4, 8, 16, 32, 63, 114, 173, 178, 65, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

A(n,k) is the number of compositions of n avoiding the pattern {1}^(k+1).

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

A(n,k) = Sum_{i=0..k} A242447(n,i).

EXAMPLE

Square array A(n,k) begins:

  1,  1,  1,  1,   1,   1,   1,   1,   1, ...

  0,  1,  1,  1,   1,   1,   1,   1,   1, ...

  0,  1,  2,  2,   2,   2,   2,   2,   2, ...

  0,  3,  3,  4,   4,   4,   4,   4,   4, ...

  0,  3,  7,  7,   8,   8,   8,   8,   8, ...

  0,  5, 11, 15,  15,  16,  16,  16,  16, ...

  0, 11, 21, 26,  31,  31,  32,  32,  32, ...

  0, 13, 34, 52,  57,  63,  63,  64,  64, ...

  0, 19, 59, 93, 114, 120, 127, 127, 128, ...

MAPLE

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,

      add(b(n-i*j, i-1, p+j, k)/j!, j=0..min(n/i, k))))

    end:

A:= (n, k)-> `if`(k>=n, `if`(n=0, 1, 2^(n-1)), b(n$2, 0, k)):

seq(seq(A(n, d-n), n=0..d), d=0..14);

MATHEMATICA

b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i<1, 0,

     Sum[b[n-i*j, i-1, p+j, k]/j!, {j, 0, Min[n/i, k]}]]];

A[n_, k_] := If[k >= n, If[n == 0, 1, 2^(n-1)], b[n, n, 0, k]];

Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Feb 02 2015, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A032020, A232432, A232464, A243082, A243083, A243084, A243085, A243086, A243087, A243088.

Main diagonal gives A011782.

A(2n,n) gives A232605.

Sequence in context: A103498 A030386 A096799 * A287847 A336201 A271369

Adjacent sequences:  A243078 A243079 A243080 * A243082 A243083 A243084

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 29 2014

STATUS

approved

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Last modified May 20 06:11 EDT 2022. Contains 353852 sequences. (Running on oeis4.)