A267981 a(n) = Catalan(n)^2*(4n + 2).

2, 6, 40, 350, 3528, 38808, 453024, 5521230, 69526600, 898283672, 11848435872, 158966514616, 2163449607200, 29802622140000, 414852500188800, 5827381213589550, 82510878636707400, 1176544010190087000, 16882265852589060000, 243611096252860135800

Offset: 0

Author

Ralf Steiner, Jan 23 2016

Keywords

• nonn
• frac

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.

Examples

			For n=3 the a(3)=350.
		

Links

• Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe (in German).

Crossrefs

Cf. A013709 (denominator). Equals twice A000891.

Programs

Magma

[Catalan(n)^2*(4*n+2):n in [0..20]]; // Vincenzo Librandi, Jan 25 2016

Mathematica

Table[CatalanNumber[n]^2 (4 n + 2), {n, 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

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

Extensions

More terms from Vincenzo Librandi, Jan 25 2016