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A234043
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a(n) = binomial(5*(n+1),4)/5, with n >= 0.
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8
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1, 42, 273, 969, 2530, 5481, 10472, 18278, 29799, 46060, 68211, 97527, 135408, 183379, 243090, 316316, 404957, 511038, 636709, 784245, 956046, 1154637, 1382668, 1642914, 1938275, 2271776, 2646567, 3065923, 3533244, 4052055, 4626006, 5258872, 5954553, 6717074
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OFFSET
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0,2
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COMMENTS
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Used as one of the 5-section parts of A234042.
The Fuss-Catalan numbers are Cat(d,k) = (1/(k*(d-1)+1))*binomial(k*d,k) and enumerate the (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link). a(n) = Cat(n,5) (Offset=1), so enumerates the (n+1)-gon partitions of a (5*(n-1)+2)-gon. Analogous series are A000326 (k=3) and A100157 (k=4). - Tom Copeland, Oct 05 2014
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LINKS
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FORMULA
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G.f: (1 + 37*x + 73*x^2 + 14*x^3)/(1-x)^5.
a(n) = (n+1)*(5*n+2)*(5*n+3)*(5*n+4)/24.
Sum_{n>=0} 1/a(n) = 10*sqrt(5)*log(phi) + 5*log(5) - 2*sqrt(25-38/sqrt(5))*Pi, where phi is the golden ratio (A001622).
Sum_{n>=0} (-1)^n/a(n) = 4*sqrt(5)*log(phi) + 2*sqrt(26-38/sqrt(5))*Pi - 32*log(2). (End)
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MATHEMATICA
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CoefficientList[Series[(1 + 37 x + 73 x^2 + 14 x^3)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
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PROG
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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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