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A267980
a(n) = Catalan(n)^2*(4n + 1).
0
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.
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
KEYWORD
nonn
AUTHOR
Ralf Steiner, Jan 23 2016
EXTENSIONS
Corrected and extended by Vincenzo Librandi, Jan 25 2016
STATUS
approved