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

0,2

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

Table of n, a(n) for n=0..18.

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

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

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