A110690 - OEIS

%I #19 Nov 05 2019 15:40:09

%S 1,22,193,1045,4180,13566,37764,93456,210705,440440,864721,1610401,

%T 2866864,4908580,8123280,13046616,20404233,31162242,46587145,68316325,

%U 98440276,139597810,195085540,268983000,366294825,493111476,656790057

%N 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 (see p. 243, H*(2,6,n)).

%H Chai Wah Wu, <a href="/A110690/b110690.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = (n+1)*(n+2)^2*(n+3)*(n+4)(31*n^3 + 236*n^2 + 545*n + 420)/20160.

%F G.f.: (1 + 13*x + 31*x^2 + 16*x^3 + x^4)/(1-x)^9. - _R. J. Mathar_, Nov 01 2015

%p a:=n->(n+1)*(n+2)^2*(n+3)*(n+4)*(31*n^3+236*n^2+545*n+420)/20160: seq(a(n),n=0..31);

%t CoefficientList[Series[(1+13*x+31*x^2+16*x^3+x^4)/(1-x)^9, {x,0,50}], x] (* _G. C. Greubel_, Sep 06 2017 *)

%t LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,22,193,1045,4180,13566,37764,93456,210705},30] (* _Harvey P. Dale_, Nov 05 2019 *)

%o (Python)

%o A110690_list, m = [], [62, -65, 20, 0, 1, 1, 1, 1, 1]

%o for _ in range(10001):

%o     A110690_list.append(m[-1])

%o     for i in range(8):

%o         m[i+1] += m[i] # _Chai Wah Wu_, Jun 12 2016

%o (PARI) x='x+O('x^50); Vec((1+13*x+31*x^2+16*x^3+x^4)/(1-x)^9) \\ _G. C. Greubel_, Sep 06 2017

%K nonn,easy

%O 0,2

%A _Emeric Deutsch_, Aug 02 2005