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A108650 a(n) = (n+1)^2*(n+2)*(n+3)*(3*n+4)/24.

%I A108650 #42 Oct 20 2023 01:48:42
%S A108650 1,14,75,260,700,1596,3234,6000,10395,17050,26741,40404,59150,84280,
%T A108650 117300,159936,214149,282150,366415,469700,595056,745844,925750,
%U A108650 1138800,1389375,1682226,2022489,2415700,2867810,3385200,3974696,4643584
%N A108650 a(n) = (n+1)^2*(n+2)*(n+3)*(3*n+4)/24.
%C A108650 Kekulé numbers for certain benzenoids.
%H A108650 Bo Gyu Jeong, <a href="/A108650/b108650.txt">Table of n, a(n) for n = 0..5000</a>
%H A108650 Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, <a href="https://arxiv.org/abs/1812.02955">Mixed restricted Stirling numbers</a>, arXiv:1812.02955 [math.CO], 2018.
%H A108650 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. 230, no. 26).
%H A108650 C. Krishnamachari, <a href="/A001296/a001296.pdf">The operator (xD)^n</a>, J. Indian Math. Soc., 15 (1923),3-4. [Annotated scanned copy]
%H A108650 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A108650 From _Zerinvary Lajos_, Jan 20 2007: (Start)
%F A108650 a(n) = A001477(n+1)*A001296(n+1) = (n+1)*A001296(n+1).
%F A108650 a(n) = (n+1)*Stirling2(n+3,n+1). (End)
%F A108650 From _Colin Barker_, Apr 22 2020: (Start)
%F A108650 G.f.: (1 + 8*x + 6*x^2) / (1 - x)^6.
%F A108650 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
%F A108650 (End)
%F A108650 From _Amiram Eldar_, May 29 2022: (Start)
%F A108650 Sum_{n>=0} 1/a(n) 2*Pi^2 + 54*sqrt(3)*Pi/5 + 486*log(3)/5 - 921/5.
%F A108650 Sum_{n>=0} (-1)^n/a(n) = Pi^2 - 108*sqrt(3)*Pi/5 - 528*log(2)/5 + 909/5. (End)
%F A108650 E.g.f.: (1/24)*(24 +312*x +576*x^2 +304*x^3 +55*x^4 +3*x^5)*exp(x). - _G. C. Greubel_, Oct 19 2023
%p A108650 a:= n-> (n+1)^2*(n+2)*(n+3)*(3*n+4)/24: seq(a(n),n=0..36);
%p A108650 seq((n+1)*stirling2(n+3,n+1), n=0..32); # _Zerinvary Lajos_, Jan 20 2007
%t A108650 Table[((n+1)^2 (n+2)(n+3)(3n+4))/24,{n,0,40}] (* or *) Table[n StirlingS2[n+2,n],{n,40}] (* _Harvey P. Dale_, Dec 01 2013 *)
%o A108650 (PARI) Vec((1 + 8*x + 6*x^2) / (1 - x)^6 + O(x^30)) \\ _Colin Barker_, Apr 22 2020
%o A108650 (Magma) [(n+1)*StirlingSecond(n+3,n+1): n in [0..40]]; // _G. C. Greubel_, Oct 19 2023
%o A108650 (SageMath) [(n+1)*stirling_number2(n+3,n+1) for n in range(41)] # _G. C. Greubel_, Oct 19 2023
%Y A108650 Cf. A108645, A108646, A108647, A108648, A108649.
%Y A108650 Cf. A001296, A001477.
%K A108650 nonn,easy
%O A108650 0,2
%A A108650 _Emeric Deutsch_, Jun 13 2005