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Expansion of (1-sqrt(1-(2*(1-sqrt(1-4*x^2)))/x))/2.
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%I #51 Jan 30 2020 21:29:17

%S 0,1,1,3,7,22,67,225,765,2704,9710,35558,131859,494892,1874901,

%T 7162807,27558511,106695148,415346144,1624780952,6383671910,

%U 25179642120,99670897534,395810459602,1576464630375,6295827843098

%N Expansion of (1-sqrt(1-(2*(1-sqrt(1-4*x^2)))/x))/2.

%C The sequence 1, 1, 3, 7, ... with offset 0 is the Riordan transform with the Riordan matrix A053121 (the inverse of the Chebyshev S matrix A049310) of the Catalan sequence A000108. - _Wolfdieter Lang_, Feb 18 2017

%F a(n) = sum(i=0..(n-1)/2, binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i))/n, n>0, a(0)=0.

%F G.f. A(x) = x*C(x^2)*C(x*C(x^2)), where C(x) is g.f. A000108.

%F G.f. A(x) satisfies A(x)=x*(1/(1-A(x))+A(x)^2-A(x)^3).

%F a(n) ~ 17^(n+1/2) / (sqrt(15*Pi) * n^(3/2) * 4^(n+1)). - _Vaclav Kotesovec_, Jun 15 2014

%F Conjecture D-finite with recurrence: 2*n*(2*n+1)*a(n) +(-49*n^2+97*n-36)*a(n-1) +12*(10*n^2-42*n+41)*a(n-2) +4*(49*n-97)*(n-3)*a(n-3) -544*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020

%t CoefficientList[Series[1/2 - Sqrt[(-2 + x + 2*Sqrt[1-4*x^2])/x]/2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 15 2014 *)

%o (Maxima)

%o a(n):=sum(binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i),i,0,(n-1)/2)/(n);

%o (PARI) a(n) = if (n, sum(i=0,(n-1)/2, binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i))/n, 0); \\ _Michel Marcus_, Jun 09 2014

%Y Cf. A000108, A049310, A053121, A101499.

%K nonn,easy

%O 0,4

%A _Vladimir Kruchinin_, Jun 09 2014