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A089155
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a(n) = (2*n)!*(Integral_{x=0..sqrt(2/3)} 1/(1-x^2)^(n+1/2) dx)/((n!*2^n)*sqrt(2)).
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1
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1, 5, 47, 687, 14001, 369645, 12013695, 463731975, 20719022625, 1051207269525, 59685242540175, 3748724456313375, 258029176261158225, 19313242781012905725, 1561734017924407502175, 135675820682608239408375
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OFFSET
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1,2
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COMMENTS
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Also numerator of I(n) = (Integral_{x=0..sqrt(2)} (1+x^2)^n dx)/sqrt(2). E.g., I(3) = 687/105. Offset is 0. The denominator is b(n) = (2*n+2)!/((n+1)!*2^(n+1)). - Al Hakanson (hawkuu(AT)excite.com), Apr 02 2004
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LINKS
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FORMULA
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a(n) ~ 3*6^n*n^n/(sqrt(2)*exp(n)). - Vaclav Kotesovec, added Sep 29 2013, simplified Nov 17 2013
D-finite with recurrence: a(n) +(-8*n+11)*a(n-1) +6*(2*n-3)*(n-2)*a(n-2)=0. - R. J. Mathar, Jan 24 2020
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MATHEMATICA
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f[n_] := Simplify[(2n)!Integrate[1/(1 - x^2)^(n + 1/2), {x, 0, Sqrt[2/3]}]/(n!2^n Sqrt[2])]; Table[ f[n], {n, 1, 16}] (* Robert G. Wilson v, Feb 27 2004 *)
With[{nn=20}, CoefficientList[Series[1/(Sqrt[1-6x](1-2x)), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Dec 17 2013 *)
Table[6^(n - 1) (n - 3/2)! HypergeometricPFQ[{1, 1 - n}, {3/2 - n},
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PROG
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(PARI) x='x+O('x^50); Vec(serlaplace(1/(sqrt(1-6*x)*(1-2*x)))) \\ G. C. Greubel, May 24 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Al Hakanson (hawkuu(AT)excite.com), Dec 21 2003
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EXTENSIONS
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STATUS
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approved
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