A089155 a(n) = (2*n)!*(Integral_{x=0..sqrt(2/3)} 1/(1-x^2)^(n+1/2) dx)/((n!*2^n)*sqrt(2)).

1, 5, 47, 687, 14001, 369645, 12013695, 463731975, 20719022625, 1051207269525, 59685242540175, 3748724456313375, 258029176261158225, 19313242781012905725, 1561734017924407502175, 135675820682608239408375

Offset

1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..345

Formula

E.g.f.: 1/(sqrt(1-6*x)*(1-2*x)). - Vladeta Jovovic, Dec 23 2003

a(n) ~ 3*6^n*n^n/(sqrt(2)*exp(n)). - Vaclav Kotesovec, added Sep 29 2013, simplified Nov 17 2013

a(n) = 6^(n-1)*(n-3/2)!*2F1(1,1-n; 3/2-n; 1/3)/sqrt(Pi). - Benedict W. J. Irwin, May 26 2016

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

Mathematica

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},
1/3]/Sqrt[Pi], {n, 1, 10}] (* Benedict W. J. Irwin, May 26 2016 *)

Prog

(PARI) x='x+O('x^50); Vec(serlaplace(1/(sqrt(1-6*x)*(1-2*x)))) \\ G. C. Greubel, May 24 2017

Crossrefs

Cf. A089252.

Sequence in context: A052802 A098799 A270529 * A254530 A086555 A246040

Adjacent sequences: A089152 A089153 A089154 * A089156 A089157 A089158

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)excite.com), Dec 21 2003

Extensions

More terms from Vladeta Jovovic, Dec 23 2003

Status

approved