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A106351
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Triangle read by rows: T(n,k) = number of compositions of n into k parts such that no two adjacent parts are equal.
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34
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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
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OFFSET
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1,5
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LINKS
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A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.
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FORMULA
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G.f.: 1/(1 - Sum_{k>0} (-1)^(k+1)*x^k*y^k/(1-x^k).
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EXAMPLE
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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;
...
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A131044 (at least two adjacent parts are equal).
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KEYWORD
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AUTHOR
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STATUS
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approved
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