A126119 -id:A126119

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A126115

E.g.f.: sqrt(1+2*x)/(1-2*x).

+10
3

1, 3, 11, 69, 537, 5475, 64755, 916965, 14536305, 263680515, 5239150875, 115916048325, 2768235849225, 72290366223075, 2016224400665475, 60700190066641125, 1936215798778886625, 66023235942444655875, 2370503834057244760875, 90300788789652000685125, 3603830757053442135845625

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Old name: Numerators of sequence of fractions with e.g.f. sqrt(1+x)/(1-x).

Denominators are successive powers of 2.

LINKS

Robert Israel, Table of n, a(n) for n = 0..403

FORMULA

b(n) = a(n)/n! satisfies b(n) = (3*b(n-1) + 2*(2*n-3)*b(n-2))/n, b(0)=1, b(1)=3. - Sergei N. Gladkovskii, Jul 22 2012, corrected by Robert Israel, Mar 12 2018

D-finite with recurrence: a(n+2) = (2*(n+1))*(1+2*n)*a(n)+3*a(n+1). - Robert Israel, Mar 12 2018

E.g.f.: sqrt(1+2*x)/(1-2*x). - Sergei N. Gladkovskii, Jul 22 2012

EXAMPLE

The fractions are 1, 3/2, 11/4, 69/8, 537/16, 5475/32, 64755/64, 916965/128, ...

MAPLE

f:= gfun:-rectoproc({-2*(n+1)*(1+2*n)*a(n)-3*a(n+1)+a(n+2), a(0)=1, a(1)=3}, a(n), remember):

map(f, [$0..30]); # Robert Israel, Mar 12 2018

MATHEMATICA

With[{nn=20}, CoefficientList[Series[Sqrt[1+2x]/(1-2x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Sep 21 2018 *)

CROSSREFS

Cf. A001147, A126119.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Mar 22 2007

EXTENSIONS

Better name by Sergei N. Gladkovskii, Jul 22 2012

Edited by Robert Israel, Mar 12 2018

STATUS

approved

A126121

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Denominators are successive powers of 2.

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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