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A295383
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a(n) = (2*n)! * [x^(2*n)] (-x/(1 - x))^n/((1 - x)*n!).
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6
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1, -4, 72, -2400, 117600, -7620480, 614718720, -59364264960, 6678479808000, -857813628672000, 123868287980236800, -19863969090648883200, 3502679882984419737600, -673592285189311488000000, 140299650258002307072000000, -31464534897861317399347200000
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: 2*K(-16*x)/Pi, where K() is the complete elliptic integral of the first kind.
a(n) = (-16)^n*Gamma(n+1/2)^2/(Pi*Gamma(n+1)).
a(n) = (-16)^n*binomial(n-1/2,-1/2)*Gamma(n+1/2)/sqrt(Pi).
a(n) ~ (-exp(-1)*n*16)^n/sqrt(n*Pi/2). (End)
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MAPLE
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a := n -> (-16)^n*GAMMA(n+1/2)^2/(Pi*GAMMA(n+1)):
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MATHEMATICA
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Table[(2 n)! SeriesCoefficient[(-x/(1 - x))^n /((1 - x) n!), {x, 0, 2 n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[2 EllipticK[-16 x]/Pi, {x, 0, nmax}], x] Range[0, nmax]!
Table[(-16)^n*Gamma[n + 1/2]^2/(Pi*Gamma[n + 1]), {n, 0, 50}] (* G. C. Greubel, Feb 06 2018 *)
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PROG
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(PARI) for(n=0, 30, print1(round((-16)^n*gamma(n+1/2)^2/(Pi*gamma(n+1))), ", ")) \\ G. C. Greubel, Feb 06 2018
(Magma) R:= RealField(); [Round((-16)^n*Gamma(n+1/2)^2/(Pi(R)*Gamma(n+1) )): n in [0..30]]; // G. C. Greubel, Feb 06 2018
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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