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A134288
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a(n) = binomial(n+7,7)*binomial(n+7,6)/(n+7).
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8
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1, 28, 336, 2520, 13860, 60984, 226512, 736164, 2147145, 5725720, 14158144, 32821152, 71954064, 150233760, 300467520, 578399976, 1075994073, 1941008916, 3405278800, 5824819000, 9735768900, 15931258200, 25565576400, 40293571500, 62455035825, 95315993136
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OFFSET
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0,2
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COMMENTS
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Seventh column of Narayana triangle A001263.
In the Narayana triangle N(n,k)= A001263(n,k) the sequence of column nr. k>=1 (without leading zeros coincides with the sequence of the diagonal d=k-1>=0 (d=0 for the main diagonal N(n,n)).
Kekulé numbers K(O(1,6,n)) for certain benzenoids (see the Cyvin-Gutman reference, p. 105, eq. (i)).
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=8. - N. J. A. Sloane, Aug 28 2010.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
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FORMULA
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O.g.f.: (1 + 15*x + 50*x^2 + 50*x^3 + 15*x^4 + x^5)/(1-x)^13. Numerator polynomial is the sixth row polynomial of the Narayana triangle.
a(n) = binomial(n+6,6)^2 - binomial(n+6,5)*binomial(n+6,7). - Gary Detlefs, Dec 05 2011
Sum_{n>=0} 1/a(n) = 1741019/20 - 8820*Pi^2.
Sum_{n>=0} (-1)^n/a(n) = 210*Pi^2 - 41433/20. (End)
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MAPLE
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a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6))^2*(n+7))/3628800:
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MATHEMATICA
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Table[Binomial[n+7, 7] Binomial[n+7, 6]/(n+7), {n, 0, 30}] (* or *) LinearRecurrence[{13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1}, {1, 28, 336, 2520, 13860, 60984, 226512, 736164, 2147145, 5725720, 14158144, 32821152, 71954064}, 30] (* Harvey P. Dale, Sep 28 2016 *)
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PROG
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(PARI) Vec((1+15*x+50*x^2+50*x^3+15*x^4+x^5)/(1-x)^13 + O(x^30)) \\ Altug Alkan, Sep 01 2016
(PARI) vector(30, n, binomial(n+6, 7)*binomial(n+5, 5)/6) \\ G. C. Greubel, Aug 27 2019
(Magma) [Binomial(n+7, 7)*Binomial(n+6, 5)/6: n in [0..30]]; // G. C. Greubel, Aug 27 2019
(Sage) [binomial(n+7, 7)*binomial(n+6, 5)/6 for n in (0..30)] # G. C. Greubel, Aug 27 2019
(GAP) List([0..30], n-> Binomial(n+7, 7)*Binomial(n+6, 5)/6); # G. C. Greubel, Aug 27 2019
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CROSSREFS
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Cf. A108679 (sixth column of Narayana triangle).
Cf. A134289 (eighth column of Narayana triangle).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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