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A107984 Triangle read by rows: T(n,k) = (k+1)*(n+2)*(2n-k+3)*(n-k+1)/6 for 0 <= k <= n.

%I A107984 #29 Jul 11 2025 08:47:08
%S A107984 1,5,4,14,16,10,30,40,35,20,55,80,81,64,35,91,140,154,140,105,56,140,
%T A107984 224,260,256,220,160,84,204,336,405,420,390,324,231,120,285,480,595,
%U A107984 640,625,560,455,320,165,385,660,836,924,935,880,770,616,429,220,506,880
%N A107984 Triangle read by rows: T(n,k) = (k+1)*(n+2)*(2n-k+3)*(n-k+1)/6 for 0 <= k <= n.
%C A107984 Kekulé numbers for certain benzenoids. Column 0 yields A000330. Main diagonal yields A000292. Row sums yield A006414.
%H A107984 S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 237, K{B(n,3,-l)}).
%F A107984 T(n-2,k-1) = n*(2*n-k)*(n-k)*k/6. - _M. F. Hasler_, Dec 26 2016
%F A107984 G.f.: (1 + x - 4*x^2*y + x^3*y^2 + x^4*y^2)/((1 - x)^4*(1 - x*y)^4). - _Stefano Spezia_, Jul 11 2025
%e A107984 Triangle begins:
%e A107984    1;
%e A107984    5,  4;
%e A107984   14, 16, 10;
%e A107984   30, 40, 35, 20;
%p A107984 T:=proc(n,k) if k<=n then (k+1)*(n+2)*(2*n-k+3)*(n-k+1)/6 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%o A107984 (PARI) A107984_row(n)=vector(n+1,k,k*(2*n-k+4)*(n-k+2))*(n+2)/6 \\ _M. F. Hasler_, Dec 26 2016
%Y A107984 Cf. A000330, A000292, A006414.
%K A107984 nonn,tabl
%O A107984 0,2
%A A107984 _Emeric Deutsch_, Jun 12 2005