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A sequence related to the even-indexed Catalan numbers.
1

%I #16 Dec 11 2019 23:42:24

%S 1,2,6,20,78,332,1516,7240,35734,180620,929940,4858328,25687052,

%T 137177016,738819672,4008435984,21886788582,120178329740,663179894788,

%U 3675923244856,20456707469540,114254175491304,640223315385576

%N A sequence related to the even-indexed Catalan numbers.

%C Binomial transform of A098465. Second binomial transform of (1,0,2,0,14,0,132,0,1430,...) (set odd-indexed Catalan numbers to zero).

%H Vincenzo Librandi, <a href="/A098469/b098469.txt">Table of n, a(n) for n = 0..300</a>

%F G.f.: (sqrt(1+2*x) - sqrt(1-6*x))/(4*x*sqrt(1-2*x)).

%F a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(k)*2^(n-2k).

%F a(n) = Sum_{k=0..n} C(n,k)*2^(n-k)*C(k)*(1-(-1)^k)/2.

%F 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

%F a(n) ~ 3*6^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012

%t 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 *)

%o (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

%Y Cf. A048990.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Sep 09 2004, corrected Mar 31 2007