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 A090470 E.g.f.: 1/((1-4*x)*sqrt(1-2*x)). 1
 1, 5, 43, 531, 8601, 172965, 4161555, 116658675, 3735104625, 134498225925, 5380583766075, 236759435017875, 11364769115001225, 590975899833644325, 33094863848730759075, 1985698021207199173875, 127084865256044709638625 (list; graph; refs; listen; history; text; internal format)
 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 Conjecture: 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 Sequence in context: A132691 A256033 A251568 * A052895 A162695 A161635 Adjacent sequences:  A090467 A090468 A090469 * A090471 A090472 A090473 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

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Last modified January 20 05:16 EST 2020. Contains 331067 sequences. (Running on oeis4.)