A090470 Expansion of e.g.f.: 1/((1-4*x)*sqrt(1-2*x)).

1, 5, 43, 531, 8601, 172965, 4161555, 116658675, 3735104625, 134498225925, 5380583766075, 236759435017875, 11364769115001225, 590975899833644325, 33094863848730759075, 1985698021207199173875, 127084865256044709638625, 8641777170070911018277125, 622208177888201070015724875

Offset: 0

Author

Al Hakanson (hawkuu(AT)excite.com), Feb 25 2004

Keywords

• nonn

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, and 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

Programs

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 *)

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 */

PARI

my(x='x+O('x^25)); Vec(serlaplace(1/((1-4*x)*sqrt(1-2*x)))) \\ Joerg Arndt, Jun 05 2025

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

Extensions

Edited and extended by Robert G. Wilson v, Feb 27 2004

Simpler definition from N. J. A. Sloane, Mar 21 2007