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A182309
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Triangle T(n,k) with 2 <= k <= floor(2(n+1)/3) gives the number of length-n binary sequences with exactly k zeros and with length two for the longest run of zeros.
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0
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1, 2, 3, 2, 4, 6, 1, 5, 12, 6, 6, 20, 18, 3, 7, 30, 40, 16, 1, 8, 42, 75, 50, 10, 9, 56, 126, 120, 45, 4, 10, 72, 196, 245, 140, 30, 1, 11, 90, 288, 448, 350, 126, 15, 12, 110, 405, 756, 756, 392, 90, 5, 13, 132, 550, 1200, 1470, 1008, 357, 50, 1, 14, 156, 726
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OFFSET
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2,2
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COMMENTS
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Triangle T(n,k) captures several well known sequences. In particular, T(n,2)=(n-1), the natural numbers; T(n,3)=(n-2)(n-3)=A002378(n-3), the "oblong" numbers; T(n,4)=(n-3)(n-4)^2/2=A002411(n-4), "pentagonal pyramidal" numbers; and also T(n,5)=(n-4)C(n-4,3)=A004320(n-6). Furthermore, row sums=A000100(n+1).
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LINKS
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FORMULA
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T(n,k) = Sum_{j=1..k/2} binomial(n-k+1,j)*binomial(n-k-j+1,k-2j) for 2 <= k <= 2(n+1)/3.
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EXAMPLE
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For n=6 and k=3, T(6,3)=12 since there are 12 binary sequences of length 6 that contain 3 zeros and that have a maximum run of zeros of length 2, namely, 011100, 101100, 110100, 011001, 101001, 110010, 010011, 100110, 100101, 001110, 001101, and 001011.
Triangle T(n,k) begins
1,
2,
3, 2,
4, 6, 1,
5, 12, 6,
6, 20, 18, 3,
7, 30, 40, 16, 1,
8, 42, 75, 50, 10,
9, 56, 126, 120, 45, 4,
10, 72, 196, 245, 140, 30, 1,
11, 90, 288, 448, 350, 126, 15,
12, 110, 405, 756, 756, 392, 90, 5,
13, 132, 550, 1200, 1470, 1008, 357, 50, 1,
14, 156, 726, 1815, 2640, 2268, 1106, 266, 21,
15, 182, 936, 2640, 4455, 4620, 2898, 1016, 161, 6,
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MAPLE
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seq(seq(sum(binomial(n-k+1, j)*binomial(n-k+1-j, k-2*j), j=1..floor(k/2)), k=2..floor(2*(n+1)/3)), n=2..20);
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MATHEMATICA
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t[n_, k_] := Sum[ Binomial[n-k+1, j]*Binomial[n-k-j+1, k-2*j], {j, 1, k/2}]; Table[t[n, k], {n, 2, 15}, {k, 2, 2*(n+1)/3}] // Flatten (* Jean-François Alcover, Jun 06 2013 *)
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CROSSREFS
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Row sums of triangle T(n,k)=A000100(n+1);
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KEYWORD
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nonn,nice,easy,tabf
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AUTHOR
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STATUS
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approved
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