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A268085
a(n) = Catalan(n)^2*n.
1
0, 1, 8, 75, 784, 8820, 104544, 1288287, 16359200, 212751396, 2821056160, 38013731756, 519227905728, 7174705330000, 100136810390400, 1409850293610375, 20002637245262400, 285732116760449700, 4106497099278420000, 59341164471850545900, 861753537765219528000
OFFSET
0,3
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
FORMULA
Sum_{n>=0} a(n)/A013709(n) = 1 - 3/Pi (see A089491).
EXAMPLE
For n=3 the a(3)= 75.
MATHEMATICA
Table[CatalanNumber[n]^2 n, {n, 0, 20}]
PROG
(Magma) [Catalan(n)^2*n: n in [0..20]]; // Vincenzo Librandi, Jan 26 2016
(PARI) a(n) = n*(binomial(2*n, n)/(n+1))^2; \\ Altug Alkan, Jan 26 2016
CROSSREFS
Sequence in context: A123003 A015578 A145600 * A231617 A094735 A067306
KEYWORD
nonn,easy
AUTHOR
Ralf Steiner, Jan 26 2016
EXTENSIONS
Corrected and extended by Vincenzo Librandi, Jan 26 2016
STATUS
approved