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A106351 Triangle read by rows: T(n,k) = number of compositions of n into k parts such that no two adjacent parts are equal. 34
1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 0, 1, 4, 7, 2, 0, 0, 1, 6, 9, 6, 1, 0, 0, 1, 6, 15, 14, 3, 0, 0, 0, 1, 8, 21, 24, 15, 2, 0, 0, 0, 1, 8, 28, 46, 30, 10, 1, 0, 0, 0, 1, 10, 35, 66, 68, 30, 4, 0, 0, 0, 0, 1, 10, 46, 100, 119, 76, 24, 2, 0, 0, 0, 0, 1, 12, 54, 138, 204, 168, 69, 14, 1, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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.

FORMULA

G.f.: 1/(1 - Sum_{k>0} (-1)^(k+1)*x^k*y^k/(1-x^k).

EXAMPLE

T(6,3) = 7 because the compositions of 6 into 3 parts with no adjacent equal parts are 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3, 1+4+1.

Triangle begins:

1;

1, 0;

1, 2,  0;

1, 2,  1,  0;

1, 4,  2,  0,  0;

1, 4,  7,  2,  0, 0;

1, 6,  9,  6,  1, 0, 0;

1, 6, 15, 14,  3, 0, 0, 0;

1, 8, 21, 24, 15, 2, 0, 0, 0;

MAPLE

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

      if n<t then 0

    elif n=0 then  `if`(t=0, 1, 0)

    else add(`if`(h=j, 0, b(n-j, j, t-1)), j=1..n)

      fi

    end:

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

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

MATHEMATICA

nn=10; CoefficientList[Series[1/(1-Sum[y x^i/(1+y x^i), {i, 1, nn}]), {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, Nov 23 2013 *)

PROG

(PARI)

gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}

for(n=1, 10, my(p=polcoeff(gf(n, y), n)); for(k=1, n, print1(polcoeff(p, k), ", ")); print); \\ Andrew Howroyd, Oct 12 2017

CROSSREFS

Row sums: A003242. Columns 3-6: A106352, A106353, A106354, A106355.

Cf. A131044 (at least two adjacent parts are equal).

T(2n,n) gives A221235.

Sequence in context: A054523 A161363 A293136 * A096800 A036586 A092928

Adjacent sequences:  A106348 A106349 A106350 * A106352 A106353 A106354

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, Apr 29 2005

STATUS

approved

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Last modified January 21 10:20 EST 2022. Contains 350476 sequences. (Running on oeis4.)