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

Author

Wolfdieter Lang, Nov 13 2007

Keywords

• nonn
• easy

Comments

Seventh column of Narayana triangle A001263.

In the Narayana triangle N(n,k) = A001263(n,k) the sequence of column no. 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).

Crossrefs

Cf. A002378.

Cf. A108679 (sixth column of Narayana triangle).

Cf. A134289 (eighth column of Narayana triangle).

Programs

GAP

List([0..30], n-> Binomial(n+7, 7)*Binomial(n+6, 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

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 *)

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

Sage

[binomial(n+7, 7)*binomial(n+6, 5)/6 for n in (0..30)] # G. C. Greubel, Aug 27 2019

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)

Extensions

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