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%I #12 Jun 23 2016 03:36:35
%S 1,2,4,10,30,96,313,1038,3512,12100,42286,149440,533076,1917000,
%T 6942660,25300034,92701746,341319648,1262176462,4685724500,
%U 17457027376,65247128392,244584468174,919319871360,3464027857900,13082491517336,49513139090512,187761168576904,713325243149880,2714644395230080,10347465825694361,39500936075084958,151005816167189208,578038682649831252,2215461387843704450,8501300598574467776,32658238309503587964
%N G.f. A(x) satisfies: x*A(x) + x^2*A(x)^2 = C(x)^2, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%C It appears that a(n) = 1 (mod 2) iff n = 2*4^k - 1 for k>=0.
%C Radius of convergence is 1/4 with A(1/4) = 2*(sqrt(2) - 1).
%H Paul D. Hanna, <a href="/A273958/b273958.txt">Table of n, a(n) for n = 1..520</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = (sqrt(1 + 4*C(x)^2) - 1)/(2*x), where C(x) = (1 - sqrt(1 - 4*x))/2.
%F (2) A(x) = (1/x) * Series_Reversion( sqrt(x^2 + x^4) - (x^2 + x^4) )^2.
%F a(n) ~ 2^(2*n - 1/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 23 2016
%e G.f.: A(x) = x + 2*x^2 + 4*x^3 + 10*x^4 + 30*x^5 + 96*x^6 + 313*x^7 + 1038*x^8 + 3512*x^9 + 12100*x^10 + 42286*x^11 + 149440*x^12 + +...
%e such that x*A(x) + x^2*A(x)^2 = C(x)^2, where
%e A(x)^2 = x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 116*x^6 + 392*x^7 + 1350*x^8 + 4696*x^9 + 16500*x^10 + 58572*x^11 + 209824*x^12 +...+ A055395(n+2)*x^n +...
%e C(x)^2 = x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 +...+ A000108(n)*x^n +...
%o (PARI) /* From A(x) = (sqrt(1 + 4*C(x)^2) - 1)/(2*x) */
%o {a(n) = my(A=x,C=(1 - sqrt(1 - 4*x +x*O(x^n)))/2); A = (sqrt(1 + 4*C^2) - 1)/(2*x); polcoeff(A,n)}
%o for(n=1,40,print1(a(n),", "))
%o (PARI) /* From A(x) = (1/x)*Series_Reversion( sqrt(x^2 + x^4) - (x^2 + x^4) )^2 */
%o {a(n) = my(A=x); A = ( serreverse( sqrt(x^2 + x^4 +x^3*O(x^n)) - (x^2 + x^4) ) )^2/x; polcoeff(A,n)}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A055395.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 10 2016
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