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URL: https://oeis.org/A134291

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A134291
Tenth column (and diagonal) of Narayana triangle A001263.
9
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
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.
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
(SageMath) [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. A134290 (ninth column of Narayana triangle).
Sequence in context: A329753 A203872 A297753 * A157096 A264467 A017771
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 13 2007
STATUS
approved