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

0,2

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

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.

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)

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 *)

Prog

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

Crossrefs

Cf. A002378.

Cf. A134289 (eighth column of Narayana triangle).

Cf. A134291 (tenth column of Narayana triangle).

Sequence in context: A293971 A341427 A024379 * A359933 A341573 A078761

Adjacent sequences: A134287 A134288 A134289 * A134291 A134292 A134293

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 13 2007

Status

approved