A134290 Ninth column (and diagonal) of Narayana triangle A001263.

1, 45, 825, 9075, 70785, 429429, 2147145, 9202050, 34763300, 118195220, 367479684, 1057896060, 2848181700, 7229999700, 17420856420, 40067969766, 88385227425, 187746398125, 385374185625, 766691800875, 1482270815025, 2791289197125, 5130235085625, 9219552907500

Offset: 0

Author

Wolfdieter Lang, Nov 13 2007

Keywords

• nonn
• easy

Comments

See a comment under A134288 on the coincidence of column and diagonal sequences.

Kekulé numbers K(O(1,8,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.

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.
• W. F. Wheatley and James Ethridge (Proposers), Comment from Alan H. Rapoport, Problem 84, Missouri Journal of Mathematical Sciences, volume 8, #2, spring 1996, pages 97-102.

Crossrefs

Cf. A002378.

Cf. A134289 (eighth column of Narayana triangle).

Cf. A134291 (tenth column of Narayana triangle).

Programs

GAP

List([0..25], n-> Binomial(n+9,9)*Binomial(n+8,7)/8); # G. C. Greubel, Aug 28 2019

Magma

[Binomial(n+9,9)*Binomial(n+8,7)/8: n in [0..25]]; // G. C. Greubel, Aug 28 2019

Maple

a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8))^2*(n+9))/14631321600:
seq(a(n), n=0..23); # Peter Luschny, Sep 01 2016

Mathematica

Table[Binomial[n+9,9]*Binomial[n+8,7]/8, {n,0,25}] (* G. C. Greubel, Aug 28 2019 *)

PARI

Vec((1+28*x+196*x^2+490*x^3+490*x^4+196*x^5+28*x^6+x^7)/(1-x)^17 + O(x^25)) \\ Altug Alkan, Sep 01 2016

PARI

vector(25, n, binomial(n+8,9)*binomial(n+7,7)/8) \\ G. C. Greubel, Aug 28 2019

Sage

[binomial(n+9,9)*binomial(n+8,7)/8 for n in (0..25)] # G. C. Greubel, Aug 28 2019

Formula

a(n) = A001263(n+9,9) = binomial(n+9,9)*binomial(n+9,8)/(n+9).

O.g.f.: P(8,x)/(1-x)^17 with the numerator polynomial P(8,x) = Sum_{k=1..8} A001263(8,k)*x^(k-1), the eighth row polynomial of the Narayana triangle: P(8,x) = 1 + 28*x + 196*x^2 + 490*x^3 + 490*x^4 + 196*x^5 + 28*x^6 + x^7.

a(n) = Product_{i=1..8} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016

From Amiram Eldar, Oct 19 2020: (Start)

Sum_{n>=0} 1/a(n) = 497925669/175 - 288288*Pi^2.

Sum_{n>=0} (-1)^n/a(n) = 580367/35 - 1680*Pi^2. (End)