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A107980 Triangle read by rows: T(n,k) = (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Kekulé numbers for certain benzenoids.
REFERENCES
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)}).
LINKS
FORMULA
T(n, k) = (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24.
T(n, 0) = A000330(n+1).
T(n, n) = A006414(n).
Sum_{k=0..n} T(n, k) = A006858(n+1).
T(n, n-1) = 5*binomial(n+4, 5) = 5*A000389(n+4). - G. C. Greubel, Dec 14 2021
EXAMPLE
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;
MAPLE
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
MATHEMATICA
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 *)
PROG
(Sage)
def A107980(n, k): return (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24
flatten([[A107980(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 14 2021
CROSSREFS
Sequence in context: A109329 A169872 A115380 * A163161 A331556 A188358
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jun 12 2005
STATUS
approved

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Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)