OFFSET
0,1
COMMENTS
Numerator of (4n+2)*(Wallis-Lambert-series-1)(n) with denominator A013709(n) convergent to 2*(1-2/Pi). Proof: Both the Wallis-Lambert-series-1=4/Pi-1 and the elliptic Euler-series=1-2/Pi are absolutely convergent series. Thus any linear combination of the terms of these series will be also absolutely convergent to the value of the linear combination of these series - in this case to 2*(1-2/Pi). Q.E.D.
LINKS
Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe (in German).
FORMULA
G.f.: (Pi-2*EllipticE(16*x))/(2*Pi*x). - Benedict W. J. Irwin, Jul 14 2016
a(n) ~ 4^(2*n+1)/(Pi*n^2). - Ilya Gutkovskiy, Jul 14 2016
Recurrence: (n+1)^2*a(n) = 4*(2*n - 1)*(2*n + 1)*a(n-1). - Vaclav Kotesovec, Jul 16 2016
Sum_{n>=0} a(n)/2^(4*n+2) = 2 - 4/Pi. - Vaclav Kotesovec, Jul 16 2016
EXAMPLE
For n=3 the a(3)=350.
MATHEMATICA
Table[CatalanNumber[n]^2 (4 n + 2), {n, 0, 20}] (* Vincenzo Librandi, Jan 25 2016 *)
PROG
(Magma) [Catalan(n)^2*(4*n+2):n in [0..20]]; // Vincenzo Librandi, Jan 25 2016
(PARI) a000108(n) = binomial(2*n, n)/(n+1)
a(n) = a000108(n)^2 * (4*n+2) \\ Felix Fröhlich, Jul 14 2016
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Ralf Steiner, Jan 23 2016
EXTENSIONS
More terms from Vincenzo Librandi, Jan 25 2016
STATUS
approved