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A107980
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Triangle read by rows: T(n,k) = (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24.
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1
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1, 5, 9, 14, 30, 40, 30, 70, 105, 125, 55, 135, 216, 280, 315, 91, 231, 385, 525, 630, 686, 140, 364, 624, 880, 1100, 1260, 1344, 204, 540, 945, 1365, 1755, 2079, 2310, 2430, 285, 765, 1360, 2000, 2625, 3185, 3640, 3960, 4125, 385, 1045, 1881, 2805, 3740, 4620, 5390, 6006, 6435, 6655
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OFFSET
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0,2
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COMMENTS
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Kekulé numbers for certain benzenoids.
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 237, K{B(n,3,l)}).
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LINKS
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FORMULA
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T(n, k) = (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24.
Sum_{k=0..n} T(n, k) = A006858(n+1).
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EXAMPLE
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Triangle begins:
1;
5, 9;
14, 30, 40;
30, 70, 105, 125;
55, 135, 216, 280, 315;
91, 231, 385, 525, 630, 686;
140, 364, 624, 880, 1100, 1260, 1344;
204, 540, 945, 1365, 1755, 2079, 2310, 2430;
285, 765, 1360, 2000, 2625, 3185, 3640, 3960, 4125;
385, 1045, 1881, 2805, 3740, 4620, 5390, 6006, 6435, 6655;
506, 1386, 2520, 3800, 5130, 6426, 7616, 8640, 9450, 10010, 10296;
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MAPLE
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T:=proc(n, k) if k<=n then 1/24*(n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3) else 0 fi end: for n from 0 to 9 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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MATHEMATICA
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T[n_, k_]:= (1/6)*(n+2)*Binomial[k+2, 2]*Binomial[2*n-k+3, 2];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 14 2021 *)
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PROG
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(Sage)
def A107980(n, k): return (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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