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%I #19 Jul 22 2022 09:42:53
%S 1,15,93,372,1141,2926,6594,13476,25509,45397,76791,124488,194649,
%T 295036,435268,627096,884697,1224987,1667953,2237004,2959341,3866346,
%U 4993990,6383260,8080605,10138401,12615435,15577408,19097457,23256696
%N a(n) = (n+1)*(n+2)*(n+3)*(19*n^3 + 111*n^2 + 200*n + 120)/720.
%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.233, # 10).
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F G.f.: (1 + 8*x + 9*x^2 + x^3)/(1-x)^7.
%F a(n) = Sum_{k=0...n} A000217(n+1-k) * (A000292(n+1) - A000292(k)). - _J. M. Bergot_, Jun 07 2017
%F a(n) = A050405(n) + A181888(n+1). - _R. J. Mathar_, Jul 22 2022
%p a:=n->(n+1)*(n+2)*(n+3)*(19*n^3+111*n^2+200*n+120)/720: seq(a(n),n=0..33);
%t Table[(n + 1) (n + 2) (n + 3) (19 n^3 + 111 n^2 + 200 n + 120)/720, {n, 0, 29}] (* or *)
%t CoefficientList[Series[(1 + 8 x + 9 x^2 + x^3)/(1 - x)^7, {x, 0, 29}], x] (* or *)
%t Table[Sum[Binomial[(n + 1 - k) + 1, 2] Apply[Subtract, Map[Binomial[# + 2, 3] &, {n + 1, k}]], {k, 0, n}], {n, 0, 29}] (* _Michael De Vlieger_, Jun 08 2017 *)
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Jun 19 2005
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