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

0,2

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.

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)

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 *)

Crossrefs

Cf. A047819, A107917, A133708.

Sequence in context: A298946 A219582 A177079 * A219370 A278674 A219468

Adjacent sequences: A107912 A107913 A107914 * A107916 A107917 A107918

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 12 2005

Status

approved