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A106356 Triangle T(n,k) 0<=k<n : Number of compositions of n with k adjacent equal parts. 7
1, 1, 1, 3, 0, 1, 4, 3, 0, 1, 7, 6, 2, 0, 1, 14, 7, 8, 2, 0, 1, 23, 20, 10, 8, 2, 0, 1, 39, 42, 22, 13, 9, 2, 0, 1, 71, 72, 58, 28, 14, 10, 2, 0, 1, 124, 141, 112, 72, 33, 16, 11, 2, 0, 1, 214, 280, 219, 150, 92, 36, 18, 12, 2, 0, 1, 378, 516, 466, 311, 189, 112, 40, 20, 13, 2, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.

EXAMPLE

T(4,1) = 3 because the compositions of 4 with 1 adjacent equal part are 1+1+2, 2+1+1, 2+2.

Triangle begins:

1;

1,   1;

3,   0,  1;

4,   3,  0, 1;

7,   6,  2, 0, 1;

14,  7,  8, 2, 0, 1;

23, 20, 10, 8, 2, 0, 1;

MAPLE

b:= proc(n, h, t) option remember;

      if n=0 then `if`(t=0, 1, 0)

    elif t<0 then 0

    else add(b(n-j, j, `if`(j=h, t-1, t)), j=1..n)

      fi

    end:

T:= (n, k)-> b(n, -1, k):

seq(seq(T(n, k), k=0..n-1), n=1..15); # Alois P. Heinz, Oct 23 2011

MATHEMATICA

b[n_, h_, t_] := b[n, h, t] = If[n == 0, If[t == 0, 1, 0], If[t<0, 0, Sum[b[n-j, j, If [j == h, t-1, t]], {j, 1, n}]]]; T[n_, k_] := b[n, -1, k]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 15}] // Flatten (* Jean-Fran├žois Alcover, Feb 20 2015, after Alois P. Heinz *)

CROSSREFS

Row sums: 2^(n-1)=A000079(n-1). Columns 0-4: A003242, A106357-A106360.

Sequence in context: A256987 A048963 A119458 * A091613 A039727 A137176

Adjacent sequences:  A106353 A106354 A106355 * A106357 A106358 A106359

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, Apr 29 2005

STATUS

approved

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Last modified May 25 10:24 EDT 2017. Contains 287026 sequences.