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Expansion of (sqrt(1-8*x^2)+8*x^2+2*x-1)/(2*x*sqrt(1-8*x^2)).
1

%I #16 Nov 19 2021 10:38:58

%S 1,2,4,4,24,16,160,80,1120,448,8064,2688,59136,16896,439296,109824,

%T 3294720,732160,24893440,4978688,189190144,34398208,1444724736,

%U 240787456,11076222976,1704034304,85201715200,12171673600,657270374400

%N Expansion of (sqrt(1-8*x^2)+8*x^2+2*x-1)/(2*x*sqrt(1-8*x^2)).

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

%F a(n) = sum{k=0..floor(n/2), 2^(n-k) * A000108(k) * C(k+1, n-k)}.

%F Conjecture D-finite with recurrence: 11*n*(n+1)*a(n)+4*n*(4*n+1)*a(n-1) +8*(27-11*n^2)*a(n-2) -32*(4*n+9)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Nov 09 2012

%F a(n) ~ 2^((3*n + 1)/2) / sqrt(Pi*n) if n is even and a(n) ~ 2^((3*n + 2)/2) / (sqrt(Pi)*n^(3/2)) if n is odd. - _Vaclav Kotesovec_, Nov 19 2021

%Y Cf. A025225, A059304.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 23 2005