OFFSET
0,2
COMMENTS
With a different offset, numerator of I(n) = ((integral_{x=0..1} 1/(1+x^2)^(n+1/2) dx * sqrt(1/2)). Denominator is b(n)=(2*n)!/(2*n!). E.g. I(3)=43/60, b(3)=60.
LINKS
Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477 [math.CO], 2014 (see page 15).
Math StackExchange, Closed form of the sequence 2F1(1/2,-n,3,1/2), Oct 18 2014
FORMULA
a(n) = ((((0+1)*4+1)*8+3)*12+15)... in which the multiplicand is 4*n and the addend is (2*n)!/(n!*2^n), with offset 1. E.g. a(3)=43
a(n) ~ n! * 2^(2*n+1/2). - Vaclav Kotesovec, Jun 26 2013
D-finite with recurrence: a(n) +(-6*n+1)*a(n-1) +4*(2*n-1)*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 13 2014
0 = a(n)*(-11520*a(n+2) - 480*a(n+3) + 7520*a(n+4) - 1780*a(n+5) + 114*a(n+6) - 2*a(n+7)) + a(n+1)*(-3744*a(n+2) - 608*a(n+3) + 2100*a(n+4) - 336*a(n+5) + 11*a(n+6)) + a(n+2)*(-576*a(n+2) - 224*a(n+3) + 246*a(n+4) - 21*a(n+5)) + a(n+3)*(-24*a(n+3) + 12*a(n+4)) for all n>=0. - Michael Somos, Oct 18 2014
0 = a(n)*(a(n+1)*(+128*a(n+2) - 100*a(n+3) + 8*a(n+4)) + a(n+2)*(+56*a(n+2) + 2*a(n+3) - 3*a(n+4)) + a(n+3)*(+3*a(n+3))) + a(n+1)*(a(n+1)*(+3*a(n+4)) + a(n+2)*(+26*a(n+2) - 6*a(n+3))) + 3*a(n+2)^3 for all n>=0. - Michael Somos, Oct 18 2014
MATHEMATICA
f[n_] := (2^(n - 1/2)(2n - 1)!!Integrate[1/(1 + x^2)^(n + 1/2), {x, 0, 1}]); Table[ f[n], {n, 1, 17}] (* Robert G. Wilson v, Feb 27 2004 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / ((1 - 4 x) Sqrt[1 - 2 x]), {x, 0, n}]]; (* Michael Somos, Oct 18 2014 *)
a[ n_] := If[ n < 0, 0, 2^n (2 n + 1)!! Hypergeometric2F1[ 1/2, -n, 3/2, 1/2]]; (* Michael Somos, Oct 18 2014 *)
a[ n_] := If[ n < 0, 0, 2^n (2 n + 1)!! Beta[ 1/2, 1/2, n + 1]/Sqrt[2] // FunctionExpand]]; (* Michael Somos, Oct 18 2014 *)
PROG
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( 1 / (1 - 4*x) / sqrt(1 - 2*x + x * O(x^n)), n))}; /* Michael Somos, Oct 18 2014 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)excite.com), Feb 25 2004
EXTENSIONS
Edited and extended by Robert G. Wilson v, Feb 27 2004
Simpler definition from N. J. A. Sloane, Mar 21 2007
STATUS
approved