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

1, 35, 490, 4116, 24696, 116424, 457380, 1557270, 4723719, 13026013, 33157124, 78835120, 176729280, 376375104, 766192176, 1498581756, 2828205765, 5168991135, 9177226366, 15870391460, 26794167400, 44253495000, 71627692500, 113794603650, 177694650315

Offset: 0

Author

Emeric Deutsch, Jun 12 2005

Keywords

• nonn
• easy

Comments

Kekulé numbers for certain benzenoids.

Partial sums of A107917. - Peter Bala, Sep 21 2007

References

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 229).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.

Crossrefs

Cf. A047819, A107917, A133708.

Programs

Maple

a:=n->(1/75)*binomial(n+4,4)*binomial(n+5,4)*binomial(n+6,4): seq(a(n),n=0..27);
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

Mathematica

a[n_] := Binomial[n + 4, 4] * Binomial[n + 5, 4] * Binomial[n + 6, 4]/75; Array[a, 25, 0] (* Amiram Eldar, May 29 2022 *)

Formula

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

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

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

From Amiram Eldar, May 29 2022: (Start)

Sum_{n>=0} 1/a(n) = 25200*Pi^2 - 248713.

Sum_{n>=0} (-1)^n/a(n) = 376003 - 430080*log(2) - 64800*zeta(3). (End)