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A245113 G.f. A(x) satisfies A(x)^2 = 1 + 4*x*A(x)^6.

%I A245113 #38 Sep 04 2025 02:08:35
%S A245113 1,2,22,340,6118,120060,2492028,53798888,1195684230,27175425004,
%T A245113 628705751828,14756641134872,350529497005532,8410852483002200,
%U A245113 203561027031883320,4963404936414528720,121810229481173225670,3006555636255509030220,74585744314812449403300,1858695101618327423328312
%N A245113 G.f. A(x) satisfies A(x)^2 = 1 + 4*x*A(x)^6.
%C A245113 Radius of convergence of g.f. A(x) is r = 1/27 where A(r) = sqrt(3/2).
%F A245113 a(n) = 4^n * binomial((6*n - 1)/2, n) / (4*n + 1).
%F A245113 G.f. A(x) satisfies A(x) = sqrt(1 + 4*x^2*A(x)^10) + 2*x*A(x)^5.
%F A245113 G.f.: A(x) = sqrt(D(4*x)), where D(x) is the g.f. of A001764. - _Werner Schulte_, Aug 10 2015
%F A245113 From _Karol A. Penson_, Mar 19 2024: (Start)
%F A245113 a(n) = 4^n*binomial(3*n+1/2,n)/(6*n+1).
%F A245113 G.f.: 3F2([1/6, 1/2, 5/6], [3/4, 5/4], 27*x).
%F A245113 G.f.: sqrt(2)*sqrt((-(sqrt(1 - 27*x) + 3*i*sqrt(3)*sqrt(x))^(1/3) + (sqrt(1 - 27*x) - 3*i*sqrt(3)*sqrt(x))^(1/3))*i)*3^(3/4)/(6*x^(1/4)), where i is the imaginary unit.
%F A245113 a(n) = Integral_{x=0...27} x^n*W(x), where W(x) = h1(x) + h2(x) + h3(x) and
%F A245113 h1(x) = 2^(2/3)*3F2([-1/12, 1/6, 5/12], [1/3, 2/3], x/27)/(4*Pi*x^(5/6));
%F A245113 h2(x) = -3F2([1/4, 1/2, 3/4], [2/3, 4/3], x/27)/(12*Pi*sqrt(x));
%F A245113 h3(x) = -2^(1/3)*3F2([7/12, 5/6, 13/12], [4/3, 5/3], x/27)/(576*Pi*x^(1/6)).
%F A245113 This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, 27). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 27. For x -> 27, W'(x) tends to -infinity. (End)
%F A245113 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^10). - _Seiichi Manyama_, Jun 20 2025
%F A245113 a(n) ~ 27^n / (4 * n^(3/2) * sqrt(2*Pi)). - _Amiram Eldar_, Sep 04 2025
%e A245113 G.f.: A(x) = 1 + 2*x + 22*x^2 + 340*x^3 + 6118*x^4 + 120060*x^5 + ...
%e A245113 where A(x)^2 = 1 + 4*x*A(x)^6:
%e A245113 A(x)^2 = 1 + 4*x + 48*x^2 + 768*x^3 + 14080*x^4 + 279552*x^5 + ...
%e A245113 A(x)^6 = 1 + 12*x + 192*x^2 + 3520*x^3 + 69888*x^4 + 1462272*x^5 + ...
%e A245113 Related series:
%e A245113 A(x)^5 = 1 + 10*x + 150*x^2 + 2660*x^3 + 51750*x^4 + 1068012*x^5 + ...
%e A245113 A(x)^10 = 1 + 20*x + 400*x^2 + 8320*x^3 + 179200*x^4 + 3969024*x^5 + ...
%e A245113 where A(x) = sqrt(1 + 4*x^2*A(x)^10) + 2*x*A(x)^5.
%p A245113 A245113:=n->4^n*binomial((6*n-1)/2,n)/(4*n+1): seq(A245113(n), n=0..30); # _Wesley Ivan Hurt_, Aug 11 2015
%t A245113 Table[4^n*Binomial[(6 n - 1)/2, n]/(4 n + 1), {n, 0, 20}] (* _Wesley Ivan Hurt_, Aug 11 2015 *)
%o A245113 (PARI) /* From A(x)^2 = 1 + 4*x*A(x)^6 : */
%o A245113 {a(n) = local(A=1+x);for(i=1,n,A=sqrt(1 + 4*x*A^6 +x*O(x^n)));polcoeff(A,n)}
%o A245113 for(n=0,20,print1(a(n),", "))
%o A245113 (PARI) {a(n) = 4^n * binomial((6*n - 1)/2, n) / (4*n + 1)}
%o A245113 for(n=0,20,print1(a(n),", "))
%o A245113 (PARI) /* From A(x) = sqrt(1 + 4*x^2*A(x)^10) + 2*x*A(x)^5 : */
%o A245113 {a(n) = local(A=1+x);for(i=1,n,A = sqrt(1 + 4*x^2*A^10 +x*O(x^n)) + 2*x*A^5);polcoeff(A,n)}
%o A245113 for(n=0,20,print1(a(n),", "))
%Y A245113 Cf. A245114.
%Y A245113 Cf. A000984, A048990, A078531, A182122, A245112.
%K A245113 nonn,easy,changed
%O A245113 0,2
%A A245113 _Paul D. Hanna_, Jul 31 2014