A134289 Eighth column (and diagonal) of Narayana triangle A001263.

1, 36, 540, 4950, 32670, 169884, 736164, 2760615, 9202050, 27810640, 77364144, 200443464, 488259720, 1126753200, 2478857040, 5226256926, 10606227291, 20796524100, 39525557500, 73018266750, 131432880150, 231003243900, 397179490500, 669161098125, 1106346348900

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(1,7,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.
• Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Crossrefs

Cf. A002378.

Cf. A134288 (seventh column of Narayana triangle).

Cf. A134290 (ninth column of Narayana triangle).

Programs

GAP

List([0..30], n-> Binomial(n+8,8)*Binomial(n+7,6)/7); # G. C. Greubel, Aug 28 2019

Magma

[Binomial(n+8,8)*Binomial(n+7,6)/7: n in [0..30]]; // G. C. Greubel, Aug 28 2019

Maple

a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7))^2*(n+8))/203212800;
seq(a(n), n=0..24); # Peter Luschny, Sep 01 2016

Mathematica

Table[(Binomial[n + 8, 8] Binomial[n + 8, 7])/(n + 8), {n, 0, 30}] (* or *) LinearRecurrence[{15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1},{1, 36, 540, 4950, 32670, 169884, 736164, 2760615, 9202050, 27810640, 77364144, 200443464, 488259720, 1126753200, 2478857040}, 30] (* Harvey P. Dale, Jul 23 2012 *)

PARI

vector(30, n, binomial(n+7,8)*binomial(n+6,6)/7) \\ G. C. Greubel, Aug 28 2019

Sage

[binomial(n+8,8)*binomial(n+7,6)/7 for n in (0..30)] # G. C. Greubel, Aug 28 2019

Formula

a(n) = A001263(n+8,8) = binomial(n+8,8)*binomial(n+8,7)/(n+8).

O.g.f.: P(7,x)/(1-x)^15 with the numerator polynomial P(7,x) = Sum_{k=1..7} A001263(7,k)*x^(k-1), the seventh row polynomial of the Narayana triangle: P(7,x) = 1 + 21*x + 105*x^2 + 175*x^3 + 105*x^4 + 21*x^5 + x^6.

For n>14: a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15). - Harvey P. Dale, Jul 23 2012

a(n) = Product_{i=1..7} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016

From Amiram Eldar, Oct 19 2020: (Start)

Sum_{n>=0} 1/a(n) = 12767346/25 - 51744*Pi^2.

Sum_{n>=0} (-1)^n/a(n) = 1192508/75 - 114688*log(2)/5. (End)