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Revision History for A182309
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
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.
(history; published version)
(history; published version)
KEYWORD
nonn,nice,easy,tabltabf
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FORMULA
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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editing
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proposed
FORMULA
T(n,k) = Sum_{j=sum(1..k/2}binomial(n-k+1,j)*binomial(n-k-j+1,k-2j),j=1..k/2) for 2 <= k <= 2(n+1)/3.
EXAMPLE
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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approved
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MATHEMATICA
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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approved
editing