OFFSET
0,2
COMMENTS
Numerator of (4n+1)*(Wallis-Lambert-series-1)(n) with denominator A013709(n) convergent to 3-8/Pi (see formula).
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 3-8/Pi. Q.E.D.
LINKS
Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe (in German).
FORMULA
Sum_{n>=0} a(n)/A013709(n) = 3 - 8/Pi.
G.f.: (3*Pi-8*EllipticE(16*x)+(2-32*x)*EllipticK(16*x))/(4*Pi*x). - Benedict W. J. Irwin, Jul 14 2016
Recurrence: (n+1)^2*(4*n - 3)*a(n) = 4*(2*n - 1)^2*(4*n + 1)*a(n-1). - Vaclav Kotesovec, Jul 16 2016
EXAMPLE
For n=3 the a(3)= 325.
MATHEMATICA
Table[CatalanNumber[n]^2 (4 n + 1), {n, 0, 20}] (* Vincenzo Librandi, Jan 25 2016 *)
PROG
(Magma) [Catalan(n)^2*(4*n + 1): n in [0..20]]; // Vincenzo Librandi, Jan 25 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Steiner, Jan 23 2016
EXTENSIONS
Corrected and extended by Vincenzo Librandi, Jan 25 2016
STATUS
approved