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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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7
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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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.
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FORMULA
| G.f. 1/(1 - sum(k>0, (-1)^(k+1)*x^k*y^k/(1-x^k)).
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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;
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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
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CROSSREFS
| Row sums: A003242. Columns 3-6: A106352, A106353, A106354, A106355.
Cf. A131044 (at least two adjacent parts are equal).
Sequence in context: A136266 A054523 A161363 * A096800 A036586 A092928
Adjacent sequences: A106348 A106349 A106350 * A106352 A106353 A106354
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KEYWORD
| nonn,tabl
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Apr 29 2005
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