A107980 Triangle read by rows: T(n,k) = (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24.

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

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

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Cf. A000330, A000389, A006414, A006858.

Sequence in context: A109329 A169872 A115380 * A163161 A331556 A188358

Adjacent sequences: A107977 A107978 A107979 * A107981 A107982 A107983

Keyword

nonn,tabl

Author

Emeric Deutsch, Jun 12 2005

Status

approved