A268085 a(n) = Catalan(n)^2*n.

0, 1, 8, 75, 784, 8820, 104544, 1288287, 16359200, 212751396, 2821056160, 38013731756, 519227905728, 7174705330000, 100136810390400, 1409850293610375, 20002637245262400, 285732116760449700, 4106497099278420000, 59341164471850545900, 861753537765219528000

Offset: 0

Author

Ralf Steiner, Jan 26 2016

Keywords

• nonn
• easy

Comments

The series whose terms are the quotients a(n)/A013709(n) is convergent to 1-3/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 1-3/Pi. Q.E.D.

Apart from inclusion of a(0) the same as A145600. - R. J. Mathar, Feb 07 2016

Examples

			For n=3 the a(3)= 75.
		

Links

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

Crossrefs

Cf. A000108, A013709.

Programs

Magma

[Catalan(n)^2*n: n in [0..20]]; // Vincenzo Librandi, Jan 26 2016

Mathematica

Table[CatalanNumber[n]^2 n, {n, 0, 20}]

PARI

a(n) = n*(binomial(2*n, n)/(n+1))^2; \\ Altug Alkan, Jan 26 2016

Formula

Sum_{n>=0} a(n)/A013709(n) = 1 - 3/Pi (see A089491).

Extensions

Corrected and extended by Vincenzo Librandi, Jan 26 2016