%I #47 Aug 30 2024 02:56:05

%S 1,2,6,16,46,128,364,1024,2902,8192,23188,65536,185420,524288,1483096,

%T 4194304,11863910,33554432,94908420,268435456,759257636,2147483648,

%U 6074027496,17179869184,48592102396,137438953472,388736403144

%N G.f. A(x) satisfies: A(x*G(x)) = G(x), where G(x) is the g.f. for A098616(n) = Pell(n+1)*Catalan(n).

%C G.f. satisfies: A(x) = x/Series_Reversion(x*G(x)), where G(x) is the g.f. for A098616 = {1*1, 2*1, 5*2, 12*5, 29*14, 70*42, 169*132, ...}.

%C Hankel transform is 2^n. - _Paul Barry_ Jan 19 2011

%H Fung Lam, <a href="/A098617/b098617.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (sqrt(1-4*x^2) + 2*x)/(1-8*x^2).

%F a(2*n+1) = 2*8^n.

%F a(n) = sum{k=0..floor((n+1)/2), (C(n,k)-C(n,k-1))*A000129(n-2k+1)}. - _Paul Barry_ Jan 19 2011

%F a(n) = 2^n*sum(j=0..n/2, binomial((n-1)/2,j)). - _Vladimir Kruchinin_, May 18 2011

%F a(n) = Sum_{k, 0<=k<=n} A201093(n,k)*2^k. - _Philippe Deléham_, Nov 27 2011

%F G.f.: 1/(1-2x/(1-x/(1+x/(1+x/(1-x/(1-x/(1+x/(1+x/(1-x/(1-... (continued fraction). - _Philippe Deléham_, Nov 27 2011

%F Recurrence: (n+6)*a(n)=256*(n+1)*a(n-6)-128*(n+3)*a(n-4)+4*(5*n+23)*a(n-2), for even n. - _Fung Lam_, Mar 31 2014

%F Recurrence: n*a(n) = 12*(n-1)*a(n-2) - 32*(n-3)*a(n-4). - _Vaclav Kotesovec_, Mar 31 2014

%F Asymptotic approximation: a(n) ~ (4/sqrt(2))^n/sqrt(2)+2^(n+1)/sqrt(2*Pi*n^3), for even n. - _Fung Lam_, Mar 31 2014

%F 0 = a(n) * (+64*a(n+1) - 8*a(n+3)) + a(n+2) * (-8*a(n+1) + a(n+3)) if n>=0. - _Michael Somos_, Apr 07 2014

%e G.f. = 1 + 2*x + 6*x^2 + 16*x^3 + 46*x^4 + 128*x^5 + 364*x^6 + 1024*x^7 + ...

%t CoefficientList[Series[(Sqrt[1-4*x^2] + 2*x)/(1-8*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 31 2014 *)

%o (PARI) a(n)=polcoeff((sqrt(1-4*x^2+x^2*O(x^n))+2*x)/(1-8*x^2),n)

%o (Maxima)

%o a(n):=2^n*sum(binomial((n-1)/2, j),j,0,n/2); /* _Vladimir Kruchinin_, May 18 2011 */

%Y Cf. A098616, A098615, A000129, A000108.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 14 2004