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

0,2

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

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

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)

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 *)

Prog

(PARI) vector(30, n, binomial(n+7, 8)*binomial(n+6, 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
(Sage) [binomial(n+8, 8)*binomial(n+7, 6)/7 for n in (0..30)] # G. C. Greubel, Aug 28 2019
(GAP) List([0..30], n-> Binomial(n+8, 8)*Binomial(n+7, 6)/7); # G. C. Greubel, Aug 28 2019

Crossrefs

Cf. A002378.

Cf. A134288 (seventh column of Narayana triangle).

Cf. A134290 (ninth column of Narayana triangle).

Sequence in context: A233101 A183616 A008657 * A329913 A200708 A186309

Adjacent sequences: A134286 A134287 A134288 * A134290 A134291 A134292

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 13 2007

Status

approved