A098469 A sequence related to the even-indexed Catalan numbers.
1, 2, 6, 20, 78, 332, 1516, 7240, 35734, 180620, 929940, 4858328, 25687052, 137177016, 738819672, 4008435984, 21886788582, 120178329740, 663179894788, 3675923244856, 20456707469540, 114254175491304, 640223315385576
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
Crossrefs
Cf. A048990.
Programs
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Mathematica
CoefficientList[Series[(Sqrt[1+2*x]-Sqrt[1-6*x])/(4*x*Sqrt[1-2*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *) -
PARI
x='x+O('x^66); Vec((sqrt(1+2*x)-sqrt(1-6*x))/(4*x*sqrt(1-2*x))) \\ Joerg Arndt, May 11 2013
Formula
G.f.: (sqrt(1+2*x) - sqrt(1-6*x))/(4*x*sqrt(1-2*x)).
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(k)*2^(n-2k).
a(n) = Sum_{k=0..n} C(n,k)*2^(n-k)*C(k)*(1-(-1)^k)/2.
Recurrence: n*(n+1)*a(n) = 4*n*(2*n-1)*a(n-1) - 4*(2*n^2 - 4*n + 3)*a(n-2) - 16*(n-2)*(2*n-3)*a(n-3) + 48*(n-3)*(n-2)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ 3*6^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012
Comments