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

Author

Emeric Deutsch, Jun 12 2005

Keywords

• nonn
• tabl

Comments

Kekulé numbers for certain benzenoids.

Examples

			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;
		

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

Crossrefs

Cf. A000330, A000389, A006414, A006858.

Programs

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 *)

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

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