A267980 a(n) = Catalan(n)^2*(4n + 1).

1, 5, 36, 325, 3332, 37044, 435600, 5337189, 67481700, 874644628, 11566330256, 155510720820, 2120180615056, 29250721730000, 407699870875200, 5733391194015525, 81260713808878500, 1159736238615942900, 16654127124851370000, 240487877070131159700

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

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

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

Cf. A013709.

Sequence in context: A341961 A278576 A286765 * A187827 A291688 A300987

Adjacent sequences: A267977 A267978 A267979 * A267981 A267982 A267983

Keyword

nonn

Author

Ralf Steiner, Jan 23 2016

Extensions

Corrected and extended by Vincenzo Librandi, Jan 25 2016

Status

approved