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A225612
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Partial sums of the binomial coefficients C(4*n,n).
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3
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1, 5, 33, 253, 2073, 17577, 152173, 1336213, 11854513, 105997793, 953658321, 8622997453, 78291531921, 713305091521, 6518037055321, 59712126248041, 548239063327621, 5043390644753269, 46475480410336709, 428936432074181109, 3964252574286355429
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OFFSET
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0,2
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COMMENTS
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Generally (for p>1), partial sums of the binomial coefficients C(p*n,n) are asymptotic to (1/(1-(p-1)^(p-1)/p^p)) * sqrt(p/(2*Pi*n*(p-1))) * (p^p/(p-1)^(p-1))^n.
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LINKS
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FORMULA
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Recurrence: 3*n*(3*n-2)*(3*n-1)*a(n) = (283*n^3 - 411*n^2 + 182*n - 24)*a(n-1) - 8*(2*n-1)*(4*n-3)*(4*n-1)*a(n-2).
a(n) ~ 2^(8*n+17/2)/(229*sqrt(Pi*n)*3^(3*n+1/2)).
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MAPLE
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MATHEMATICA
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Table[Sum[Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]
Accumulate[Table[Binomial[4n, n], {n, 0, 20}]] (* Harvey P. Dale, Feb 01 2015 *)
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PROG
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(PARI) for(n=0, 50, print1(sum(k=0, n, binomial(4*k, k)), ", ")) \\ G. C. Greubel, Apr 01 2017
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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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