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

Author

N. J. A. Sloane, Mar 22 2007

Keywords

• nonn
• frac

Comments

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

Denominators are successive powers of 2.

Examples

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

Links

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

Crossrefs

Cf. A001147, A126119.

Programs

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 *)

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

Extensions

Better name by Sergei N. Gladkovskii, Jul 22 2012

Edited by Robert Israel, Mar 12 2018