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A267981 a(n) = Catalan(n)^2*(4n + 2). 1
2, 6, 40, 350, 3528, 38808, 453024, 5521230, 69526600, 898283672, 11848435872, 158966514616, 2163449607200, 29802622140000, 414852500188800, 5827381213589550, 82510878636707400, 1176544010190087000, 16882265852589060000, 243611096252860135800 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..19.

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

Cf. A013709 (denominator). Equals twice A000891.

Sequence in context: A120592 A277476 A277483 * A343846 A318006 A292407

Adjacent sequences:  A267978 A267979 A267980 * A267982 A267983 A267984

KEYWORD

nonn,frac

AUTHOR

Ralf Steiner, Jan 23 2016

EXTENSIONS

More terms from Vincenzo Librandi, Jan 25 2016

STATUS

approved

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Last modified September 26 16:15 EDT 2021. Contains 347670 sequences. (Running on oeis4.)