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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. 9
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 20 23:20 EST 2019. Contains 319343 sequences. (Running on oeis4.)