A134291 Tenth column (and diagonal) of Narayana triangle A001263.

1, 55, 1210, 15730, 143143, 1002001, 5725720, 27810640, 118195220, 449141836, 1551580888, 4936848280, 14620666060, 40648664980, 106847919376, 267119798440, 638337753625, 1464421905375, 3237143159250, 6917263803450

Offset

0,2

Comments

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

Kekulé numbers K(O(1,9,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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = A001263(n+10,10) = binomial(n+10,10)*binomial(n+10,9)/(n+10).

O.g.f.: P(9,x)/(1-x)^19 with the numerator polynomial P(9,x) = Sum_{k=1..9} A001263(9,k)*x^(k-1), the ninth row polynomial of the Narayana triangle: P(9,x) = 1 + 36*x + 336*x^2 + 1176*x^3 + 1764*x^4 + 1176*x^5 + 336*x^6 + 36*x^7 + x^8.

a(n) = Product_{i=1..9} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016

From Amiram Eldar, Oct 19 2020: (Start)

Sum_{n>=0} 1/a(n) = 2987553139/196 - 1544400*Pi^2.

Sum_{n>=0} (-1)^n/a(n) = 1179648*log(2)/7 - 114472793/980. (End)

Maple

a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9))^2*(n+10))/ 1316818944000:
seq(a(n), n=0..19); # Peter Luschny, Sep 01 2016

Mathematica

Table[Binomial[n + 10, 10]*Binomial[n + 10, 9]/(n + 10), {n, 0, 30}] (* Wesley Ivan Hurt, Apr 25 2017 *)

Prog

(PARI) vector(30, n, binomial(n+9, 10)*binomial(n+8, 8)/9) \\ G. C. Greubel, Aug 28 2019
(Magma) [Binomial(n+10, 10)*Binomial(n+9, 8)/9: n in [0..30]]; // G. C. Greubel, Aug 28 2019
(Sage) [binomial(n+10, 10)*binomial(n+9, 8)/9 for n in (0..30)] # G. C. Greubel, Aug 28 2019
(GAP) List([0..30], n-> Binomial(n+10, 10)*Binomial(n+9, 8)/9); # G. C. Greubel, Aug 28 2019

Crossrefs

Cf. A001263, A002378, A134288.

Cf. A134290 (ninth column of Narayana triangle).

Sequence in context: A329753 A203872 A297753 * A157096 A264467 A017771

Adjacent sequences: A134288 A134289 A134290 * A134292 A134293 A134294

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 13 2007

Status

approved