A126115 -id:A126115 - cp's OEIS frontend

A126119 Numerators of sequence of fractions with e.g.f. (1+x)/(1-x)^(3/2).

1, 5, 27, 195, 1785, 19845, 259875, 3918915, 66891825, 1274998725, 26843892075, 618718975875, 15495473018025, 419010239773125, 12167108660581875, 377607284571391875, 12473420957563190625, 436953531082636693125, 16179945969799083346875, 631461179013117650071875

Offset: 0

N. J. A. Sloane, Mar 22 2007

Denominators are successive powers of 2.

			The fractions are 1, 5/2, 27/4, 195/8, 1785/16, 19845/32, 259875/64, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001174, A126115.

CoefficientList[Series[(1+2*x)/(1-2*x)^(3/2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 25 2014 *)

{a(n)= if(n<0, 0, n!* polcoeff( (1+2*x)/ (1-2*x +x*O(x^n))^(3/2), n))} /* Michael Somos, Apr 08 2007 */

E.g.f.: (1+2*x)/(1-2*x)^(3/2)
From Sergei N. Gladkovskii, Jul 23 2012: (Start)
a(n) = (4*n+1)*((2*n)!)/( n!*(2^n) ).
G.f.: 3*Q(0), where Q(k)= 1 - (2*k+2)/(3*(2*k+1) - 9*x*(2*k+1)^2*(2*k+3)/(3*x*(2*k+1)*(2*k+3) - (2*k+2)/Q(k+1))); (continued fraction, 3rd kind, 3-step).
(End).
a(n) ~ 2^(n+5/2) * n^(n+1) / exp(n). - Vaclav Kotesovec, Feb 25 2014

A126121 Numerators of sequence of fractions with e.g.f. sqrt(1+x)/(1-x)^2.

Original entry on oeis.org

1, 5, 31, 255, 2577, 31245, 439695, 7072695, 127699425, 2562270165, 56484554175, 1358576240175, 35374065613425, 992016072172125, 29792674421484975, 954480422711190375, 32479589325536978625, 1170329273010701929125, 44502357662442514209375, 1781390379962467540641375

Offset: 0

N. J. A. Sloane, Mar 22 2007

Denominators are successive powers of 2.

			The fractions are 1, 5/2, 31/4, 255/8, 2577/16, 31245/32, 439695/64, ...

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A003148, A001174, A126115, A126119.

With[{nn=20},Numerator[CoefficientList[Series[Sqrt[1+x]/(1-x)^2,{x, 0, nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Jan 29 2016 *)

x='x+O('x^25); Vec(serlaplace(sqrt(1+2*x)/(1-2*x)^2)) \\ G. C. Greubel, May 25 2017

E.g.f.: 1/G(0) where G(k) = 1 - 4*x/(1 + x/(1 - x - (2*k+1)/( 2*k+1 - 4*(k+1)*x/G(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Jul 28 2012
From Benedict W. J. Irwin, May 19 2016: (Start)
E.g.f.: sqrt(1+2*x)/(1-2*x)^2.
a(n) = (-1)^(n+1)*2^(n-1)*(n-3/2)!*2F1(2,-n;(3/2)-n;-1)/sqrt(Pi).
(End)
D-finite with recurrence a(n) -5*a(n-1) -2*(2*n-1)*(n-1)*a(n-2)=0. - R. J. Mathar, Feb 08 2021

cp's OEIS Frontend

A126119 Numerators of sequence of fractions with e.g.f. (1+x)/(1-x)^(3/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula