A267987 a(n) = Catalan(n)^2*(4n + 4).

4, 8, 48, 400, 3920, 42336, 487872, 5889312, 73616400, 945561760, 12412647104, 165878102208, 2249987591488, 30906422960000, 429157758816000, 6015361252737600, 85011208292365200, 1210159553338375200, 17338543308064440000, 249857534618318088000

Offset: 0

Author

Ralf Steiner, Jan 23 2016

Keywords

• nonn
• easy
• frac

Comments

Numerator of the modified (4n+4) Wallis-Lambert series with denominator A013709 convergent to 4/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: 4/Pi. Q.E.D.

Crossrefs

Cf. A013709 (denominator).

Programs

Magma

[Catalan(n)^2*(4*n+4):n in [0..30]]; // Vincenzo Librandi, Jan 24 2016

Mathematica

Table[CatalanNumber[n]^2 (4 n + 4), {n, 0, 20}] (* Vincenzo Librandi, Jan 24 2016 *)

Extensions

More terms from Vincenzo Librandi, Jan 24 2016