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%I #36 Jan 30 2020 21:29:14
%S 0,1,4,16,64,260,1072,4480,18944,80928,348800,1515008,6625280,
%T 29147456,128918272,572928000,2557100032,11457170944,51514963968,
%U 232370167808,1051235287040,4768568354816,21684663148544,98835356778496,451433970008064
%N Expansion of (1 - 4*x - (1-2*x)*sqrt(1-4*x-4*x^2))/(8*x^3).
%H D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.4153/CJM-1997-015-x">On some alternative characterizations of Riordan arrays</a>, Canad. J. Math., 49 (1997), 301-320.
%F Binomial transform is A065096. - _Paul Barry_, Sep 16 2006
%F a(n) = (1/Pi)*Integral_{x=2-2*sqrt(2)..2+2*sqrt(2)} x^n*sqrt(-x^2+4x+4)*(x-2)/8. - _Paul Barry_, Sep 16 2006
%F a(n) = Sum_{k=0..n} 2^(n-k)*binomial(n,k)*2^((k-1)/2)*C((k-1)/2+1)*(1-(-1)^k)/2, where C(n)=A000108(n). - _Paul Barry_, Sep 16 2006
%F D-finite with recurrence: a(n) = (1/(n+3))*((6*n+8)*a(n-1) - (4*n-4)*a(n-2) - (8*n-16)*a(n-3)) for n > 2, with a(0)=0, a(1)=1, a(2)=4. - _Tani Akinari_, Jul 04 2013
%F a(n) ~ 2^(n + 1/4) * (1 + sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 03 2019
%t CoefficientList[Series[(1-4x-(1-2x)Sqrt[1-4x-4x^2])/(8x^3),{x,0,30}],x] (* _Harvey P. Dale_, Aug 09 2016 *)
%t Table[Sum[2^(n-k) * Binomial[n, k] * 2^((k-1)/2) * CatalanNumber[(k+1)/2] * (1 - (-1)^k)/2, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 03 2019 *)
%Y Cf. A000108, A065096.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jun 12 2002
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