A107983 Triangle read by rows: T(n,k) = (k+1)(n+2)(n+3)(n-k+2)(n-k+1)/12 for 0<=k<=n.

1, 6, 4, 20, 20, 10, 50, 60, 45, 20, 105, 140, 126, 84, 35, 196, 280, 280, 224, 140, 56, 336, 504, 540, 480, 360, 216, 84, 540, 840, 945, 900, 750, 540, 315, 120, 825, 1320, 1540, 1540, 1375, 1100, 770, 440, 165, 1210, 1980, 2376, 2464, 2310, 1980, 1540

Offset: 0

Author

Emeric Deutsch, Jun 12 2005

Keywords

• nonn
• tabl

Comments

Kekulé numbers for certain benzenoids. Column 0 yields A002415. Main diagonal yields A000292. Row sums yield A006542.

T(n,k) = number of Dyck (n+4)-paths with 4 peaks (UDs) and last descent of length k+1. For example, T(1,1)=4 counts UUDUDUDUDD, UDUUDUDUDD, UDUDUUDUDD, UDUDUDUUDD. - David Callan, Jun 26 2006

Examples

			Triangle begins:
1;
6,4;
20,20,10;
50,60,45,20;
		

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{F(n,3,-l)}).

Crossrefs

Cf. A002415, A000292, A006542.

Programs

Maple

T:=proc(n,k) if k<=n then (k+1)*(n+2)*(n+3)*(n-k+2)*(n-k+1)/12 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

Flatten[Table[((k+1)(n+2)(n+3)(n-k+2)(n-k+1))/12,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Aug 08 2013 *)