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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 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 A359290 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 February 6 14:22 EST 2023. Contains 360110 sequences. (Running on oeis4.)