A134288 a(n) = binomial(n+7,7)*binomial(n+7,6)/(n+7).

1, 28, 336, 2520, 13860, 60984, 226512, 736164, 2147145, 5725720, 14158144, 32821152, 71954064, 150233760, 300467520, 578399976, 1075994073, 1941008916, 3405278800, 5824819000, 9735768900, 15931258200, 25565576400, 40293571500, 62455035825, 95315993136

Offset

0,2

Comments

Seventh column of Narayana triangle A001263.

In the Narayana triangle N(n,k)= A001263(n,k) the sequence of column nr. k>=1 (without leading zeros coincides with the sequence of the diagonal d=k-1>=0 (d=0 for the main diagonal N(n,n)).

Kekulé numbers K(O(1,6,n)) for certain benzenoids (see the Cyvin-Gutman reference, p. 105, eq. (i)).

References

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

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=8. - N. J. A. Sloane, Aug 28 2010.

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

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.

Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Formula

a(n) = A001263(n+7,7).

O.g.f.: (1 + 15*x + 50*x^2 + 50*x^3 + 15*x^4 + x^5)/(1-x)^13. Numerator polynomial is the sixth row polynomial of the Narayana triangle.

a(n) = binomial(n+6,6)^2 - binomial(n+6,5)*binomial(n+6,7). - Gary Detlefs, Dec 05 2011

a(n) = Product_{i=1..6} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016

From Amiram Eldar, Oct 19 2020: (Start)

Sum_{n>=0} 1/a(n) = 1741019/20 - 8820*Pi^2.

Sum_{n>=0} (-1)^n/a(n) = 210*Pi^2 - 41433/20. (End)

Maple

a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6))^2*(n+7))/3628800:
seq(a(n), n=0..25); # Peter Luschny, Sep 01 2016

Mathematica

Table[Binomial[n+7, 7] Binomial[n+7, 6]/(n+7), {n, 0, 30}] (* or *) LinearRecurrence[{13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1}, {1, 28, 336, 2520, 13860, 60984, 226512, 736164, 2147145, 5725720, 14158144, 32821152, 71954064}, 30] (* Harvey P. Dale, Sep 28 2016 *)

Prog

(PARI) Vec((1+15*x+50*x^2+50*x^3+15*x^4+x^5)/(1-x)^13 + O(x^30)) \\ Altug Alkan, Sep 01 2016
(PARI) vector(30, n, binomial(n+6, 7)*binomial(n+5, 5)/6) \\ G. C. Greubel, Aug 27 2019
(Magma) [Binomial(n+7, 7)*Binomial(n+6, 5)/6: n in [0..30]]; // G. C. Greubel, Aug 27 2019
(Sage) [binomial(n+7, 7)*binomial(n+6, 5)/6 for n in (0..30)] # G. C. Greubel, Aug 27 2019
(GAP) List([0..30], n-> Binomial(n+7, 7)*Binomial(n+6, 5)/6); # G. C. Greubel, Aug 27 2019

Crossrefs

Cf. A002378.

Cf. A108679 (sixth column of Narayana triangle).

Cf. A134289 (eighth column of Narayana triangle).

Sequence in context: A159520 A027820 A092713 * A200968 A285739 A010833

Adjacent sequences: A134285 A134286 A134287 * A134289 A134290 A134291

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 13 2007

Extensions

Edited by N. J. A. Sloane, Aug 28 2010

Status

approved