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A134290 Ninth column (and diagonal) of Narayana triangle A001263. 7
1, 45, 825, 9075, 70785, 429429, 2147145, 9202050, 34763300, 118195220, 367479684, 1057896060, 2848181700, 7229999700, 17420856420, 40067969766, 88385227425, 187746398125, 385374185625, 766691800875, 1482270815025, 2791289197125, 5130235085625, 9219552907500 (list; graph; refs; listen; history; text; internal format)
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

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: A264138 A293971 A024379 * A078761 A034967 A199352

Adjacent sequences:  A134287 A134288 A134289 * A134291 A134292 A134293

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 13 2007

STATUS

approved

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Last modified February 19 19:30 EST 2020. Contains 332047 sequences. (Running on oeis4.)