cp's OEIS Frontend

A107915 a(n) = binomial(n+4,4)*binomial(n+5,4)*binomial(n+6,4)/75.

%I A107915 #29 Jan 15 2025 10:58:37
%S A107915 1,35,490,4116,24696,116424,457380,1557270,4723719,13026013,33157124,
%T A107915 78835120,176729280,376375104,766192176,1498581756,2828205765,
%U A107915 5168991135,9177226366,15870391460,26794167400,44253495000,71627692500,113794603650,177694650315
%N A107915 a(n) = binomial(n+4,4)*binomial(n+5,4)*binomial(n+6,4)/75.
%C A107915 Kekulé numbers for certain benzenoids.
%C A107915 Partial sums of A107917. - _Peter Bala_, Sep 21 2007
%D A107915 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 229).
%H A107915 T. D. Noe, <a href="/A107915/b107915.txt">Table of n, a(n) for n = 0..1000</a>
%H A107915 Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, <a href="https://doi.org/10.1016/j.jcta.2010.03.017">Bijections for Baxter families and related objects</a>, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
%H A107915 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 25.
%F A107915 a(n) = C(n,n-2)*C(n+1,n-3)*C(n+2,n-4)/(5*3!), n>=4. - _Zerinvary Lajos_, May 29 2007
%F A107915 a(n-3) = (1/144) * Sum_{1 <= x_1, x_2, x_3 <= n} x_1*x_2*x_3*(det V(x_1,x_2,x_3))^2 = 1/144*sum {1 <= i,j,k <= n} i*j*k*((i-j)(i-k)(j-k))^2, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. - _Peter Bala_, Sep 21 2007
%F A107915 G.f.: -(x^6+22*x^5+113*x^4+190*x^3+113*x^2+22*x+1)/(x-1)^13. - _Colin Barker_, Jun 06 2012
%F A107915 From _Amiram Eldar_, May 29 2022: (Start)
%F A107915 Sum_{n>=0} 1/a(n) = 25200*Pi^2 - 248713.
%F A107915 Sum_{n>=0} (-1)^n/a(n) = 376003 - 430080*log(2) - 64800*zeta(3). (End)
%p A107915 a:=n->(1/75)*binomial(n+4,4)*binomial(n+5,4)*binomial(n+6,4): seq(a(n),n=0..27);
%p A107915 seq(binomial(n,n-2)*binomial(n+1,n-3)*binomial(n+2,n-4)/(5*3!), n=4..22); # _Zerinvary Lajos_, May 29 2007
%t A107915 a[n_] := Binomial[n + 4, 4] * Binomial[n + 5, 4] * Binomial[n + 6, 4]/75; Array[a, 25, 0] (* _Amiram Eldar_, May 29 2022 *)
%Y A107915 Cf. A047819, A107917, A133708.
%K A107915 nonn,easy
%O A107915 0,2
%A A107915 _Emeric Deutsch_, Jun 12 2005