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A098469 A sequence related to the even-indexed Catalan numbers.

%I A098469 #16 Dec 11 2019 23:42:24
%S A098469 1,2,6,20,78,332,1516,7240,35734,180620,929940,4858328,25687052,
%T A098469 137177016,738819672,4008435984,21886788582,120178329740,663179894788,
%U A098469 3675923244856,20456707469540,114254175491304,640223315385576
%N A098469 A sequence related to the even-indexed Catalan numbers.
%C A098469 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 A098469 Vincenzo Librandi, <a href="/A098469/b098469.txt">Table of n, a(n) for n = 0..300</a>
%F A098469 G.f.: (sqrt(1+2*x) - sqrt(1-6*x))/(4*x*sqrt(1-2*x)).
%F A098469 a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(k)*2^(n-2k).
%F A098469 a(n) = Sum_{k=0..n} C(n,k)*2^(n-k)*C(k)*(1-(-1)^k)/2.
%F A098469 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 A098469 a(n) ~ 3*6^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012
%t A098469 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 A098469 (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 A098469 Cf. A048990.
%K A098469 easy,nonn
%O A098469 0,2
%A A098469 _Paul Barry_, Sep 09 2004, corrected Mar 31 2007