A285008 Numerator of (3/4)^n * binomial(2*n,n).

1, 3, 27, 135, 2835, 15309, 168399, 938223, 42220035, 239246865, 2727414261, 15620645313, 359274842199, 2072739474225, 23984556773175, 139110429284415, 12937269923450595, 75340571907153465, 878973338916790425, 5135054769461249325, 120160281605393234205

Offset

0,2

Comments

By analytic continuation to the entire complex plane there exist regularized values for divergent sums:

Sum_{k>=0} a(k)/A046161(k) = -i/sqrt(2).

Sum_{k>=0} (-1)^k*a(k)/A046161(k) = 1/2.

Sum_{k>=0} (-1)^(k+1)*a(k)/A046161(k) = -1/2.

Links

G. C. Greubel, Table of n, a(n) for n = 0..925

Formula

a(n) = numerator of (-3)^n*sqrt(Pi)/(Gamma(1/2-n)*Gamma(1+n)).

From Robert Israel, Apr 07 2017: (Start)

a(n) = 3*(2*n-1)*a(n-1)/A000265(n) for n >= 1.

a(n) = 3^n*binomial(2n,n)/A001316(n). (End)

Maple

A[0]:= 1:
for n from 1 to 100 do A[n]:=3*(2*n-1)*2^padic:-ordp(n, 2)/n*A[n-1] od:
seq(A[i], i=0..100); # Robert Israel, Apr 07 2017

Mathematica

Numerator[Table[(-3)^n*Sqrt[Pi]/(Gamma[1/2-n]*Gamma[1+n]), {n, 0, 20}]]

Prog

(PARI) for(n=0, 10, print1(numerator((3/4)^n*binomial(2*n, n)), ", ")) \\ G. C. Greubel, Jun 06 2017

Crossrefs

Cf. A046161 (denominators).

Cf. A000265, A001316.

Sequence in context: A034200 A306442 A080424 * A001796 A174613 A341566

Adjacent sequences: A285005 A285006 A285007 * A285009 A285010 A285011

Keyword

nonn,frac

Author

Ralf Steiner, Apr 07 2017

Status

approved