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

Author

N. J. A. Sloane, Mar 22 2007

Keywords

• nonn
• frac

Comments

Denominators are successive powers of 2.

Examples

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

Links

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

Crossrefs

Cf. A001174, A126115.

Programs

Mathematica

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

PARI

{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 */

Formula

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