A126115 E.g.f.: sqrt(1+2*x)/(1-2*x).

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

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.

Sequence in context: A342370 A345094 A074504 * A342057 A018193 A121945

Adjacent sequences: A126112 A126113 A126114 * A126116 A126117 A126118

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