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%I #11 Dec 17 2021 20:56:06
%S 1,5,9,14,30,40,30,70,105,125,55,135,216,280,315,91,231,385,525,630,
%T 686,140,364,624,880,1100,1260,1344,204,540,945,1365,1755,2079,2310,
%U 2430,285,765,1360,2000,2625,3185,3640,3960,4125,385,1045,1881,2805,3740,4620,5390,6006,6435,6655
%N Triangle read by rows: T(n,k) = (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24.
%C Kekulé numbers for certain benzenoids.
%D 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)}).
%H G. C. Greubel, <a href="/A107980/b107980.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24.
%F T(n, 0) = A000330(n+1).
%F T(n, n) = A006414(n).
%F Sum_{k=0..n} T(n, k) = A006858(n+1).
%F T(n, n-1) = 5*binomial(n+4, 5) = 5*A000389(n+4). - _G. C. Greubel_, Dec 14 2021
%e Triangle begins:
%e 1;
%e 5, 9;
%e 14, 30, 40;
%e 30, 70, 105, 125;
%e 55, 135, 216, 280, 315;
%e 91, 231, 385, 525, 630, 686;
%e 140, 364, 624, 880, 1100, 1260, 1344;
%e 204, 540, 945, 1365, 1755, 2079, 2310, 2430;
%e 285, 765, 1360, 2000, 2625, 3185, 3640, 3960, 4125;
%e 385, 1045, 1881, 2805, 3740, 4620, 5390, 6006, 6435, 6655;
%e 506, 1386, 2520, 3800, 5130, 6426, 7616, 8640, 9450, 10010, 10296;
%p 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
%t T[n_, k_]:= (1/6)*(n+2)*Binomial[k+2, 2]*Binomial[2*n-k+3, 2];
%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 14 2021 *)
%o (Sage)
%o def A107980(n,k): return (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24
%o flatten([[A107980(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Dec 14 2021
%Y Cf. A000330, A000389, A006414, A006858.
%K nonn,tabl
%O 0,2
%A _Emeric Deutsch_, Jun 12 2005
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