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A317135 G.f.: Sum_{n>=0} binomial(4*n+2, n)/(2*n+1) * x^(n+1)/(1+x)^(2*n+1).

%I A317135 #8 Jul 22 2018 03:41:54
%S A317135 1,1,4,18,92,504,2897,17235,105233,655687,4152461,26650675,172961053,
%T A317135 1133152365,7484233381,49780558057,333155274455,2241793462963,
%U A317135 15158132783457,102938343190333,701783155862751,4801340686195787,32954688422181493,226853921031126233,1565828056187052419,10834714749540715871,75142241780769151970,522240807596491906516
%N A317135 G.f.: Sum_{n>=0} binomial(4*n+2, n)/(2*n+1) * x^(n+1)/(1+x)^(2*n+1).
%C A317135 Equals the self-convolution square-root of A317134.
%F A317135 G.f. A(x) satisfies:
%F A317135 (1) A(x) = ( (1 + x*A(x)^2)^2 + sqrt( (1 + x*A(x)^2)^4 - 4*x*A(x)^2 ) ) / 2.
%F A317135 (2) A(x) = sqrt( (1/x) * Series_Reversion( 4*x/((1+x)^2 + sqrt( (1+x)^4 - 4*x ))^2 ) ).
%F A317135 (3) A(x) = Sum_{n>=0} binomial(4*n+2, n)/(2*n+1) * x^(n+1)/(1+x)^(2*n+1).
%F A317135 a(n) ~ 37^(1/4) * (101 + 16*sqrt(37))^(n + 1/2) / (4*sqrt(Pi) * n^(3/2) * 3^(3*n + 5/2)). - _Vaclav Kotesovec_, Jul 22 2018
%e A317135 G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 92*x^4 + 504*x^5 + 2897*x^6 + 17235*x^7 + 105233*x^8 + 655687*x^9 + 4152461*x^10 + ...
%e A317135 such that
%e A317135 A(x) = 1/(1+x) + 2*x/(1+x)^3 + 9*x^2/(1+x)^5 + 52*x^3/(1+x)^7 + 340*x^4/(1+x)^9 + 2394*x^5/(1+x)^11 + ... + A069271(n)*x^n/(1+x)^(2*n+1) + ...
%t A317135 CoefficientList[Sqrt[1/x * InverseSeries[Series[4*x/((1 + x)^2 + Sqrt[(1 + x)^4 - 4*x])^2, {x, 0, 30}], x]], x] (* _Vaclav Kotesovec_, Jul 22 2018 *)
%o A317135 (PARI) {a(n) = my(A = sum(m=0, n, binomial(4*m+2, m)/(2*m+1) * x^m / (1+x +x*O(x^n))^(2*m+1))); polcoeff(A, n)}
%o A317135 for(n=0, 30, print1(a(n), ", "))
%o A317135 (PARI) {a(n) = my(A = sqrt( (1/x) * serreverse( 4*x/((1+x)^2 + sqrt( (1+x)^4 - 4*x  + x*O(x^n)))^2 ))); polcoeff(A, n)}
%o A317135 for(n=0, 30, print1(a(n), ", "))
%Y A317135 Cf. A317134.
%K A317135 nonn
%O A317135 0,3
%A A317135 _Paul D. Hanna_, Jul 22 2018