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A108648
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a(n) = (n+1)^2*(n+2)^3*(n+3)/24.
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5
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1, 18, 120, 500, 1575, 4116, 9408, 19440, 37125, 66550, 113256, 184548, 289835, 441000, 652800, 943296, 1334313, 1851930, 2527000, 3395700, 4500111, 5888828, 7617600, 9750000, 12358125, 15523326, 19336968, 23901220, 29329875, 35749200
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OFFSET
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0,2
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COMMENTS
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Kekulé numbers for certain benzenoids.
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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 (p. 230, no. 24).
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LINKS
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FORMULA
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G.f.: (1 + 11*x + 15*x^2 + 3*x^3) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
(End)
Sum_{n>=0} 1/a(n) = 24*zeta(3) + 6*Pi^2 - 87.
Sum_{n>=0} (-1)^n/a(n) = 99 - Pi^2 - 96*log(2) - 18*zeta(3). (End)
E.g.f.: (24 + 408*x + 1020*x^2 + 772*x^3 + 224*x^4 + 26*x^5 + x^6)*exp(x)/4!. - G. C. Greubel, Oct 28 2022
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MAPLE
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a:=(n+1)^2*(n+2)^3*(n+3)/24: seq(a(n), n=0..36);
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MATHEMATICA
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Table[(n+1)^2*(n+2)^3*(n+3)/24, {n, 0, 30}] (* G. C. Greubel, Oct 28 2022 *)
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PROG
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(PARI) Vec((1 + 11*x + 15*x^2 + 3*x^3) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Apr 22 2020
(Magma) [(n+1)^2*(n+2)^3*(n+3)/24: n in [0..30]]; // G. C. Greubel, Oct 28 2022
(SageMath) [(n+1)^2*(n+2)^3*(n+3)/24 for n in (0..30)] # G. C. Greubel, Oct 28 2022
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CROSSREFS
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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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