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A134289
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Eighth column (and diagonal) of Narayana triangle A001263.
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7
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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
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OFFSET
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0,2
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COMMENTS
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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)).
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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.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
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FORMULA
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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
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)
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MAPLE
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a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7))^2*(n+8))/203212800;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A134288 (seventh column of Narayana triangle).
Cf. A134290 (ninth column of Narayana triangle).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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