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A245112 G.f. A(x) satisfies A(x)^2 = 1 + 4*x*A(x)^5.

%I A245112 #33 Sep 02 2025 04:06:40
%S A245112 1,2,18,224,3230,50688,840420,14483456,256856886,4656988160,
%T A245112 85929839996,1608379269120,30463651429484,582796191989760,
%U A245112 11245047027447240,218581150665277440,4276257634911525670,84135742205488791552,1663738200769421021580,33047906167191995678720
%N A245112 G.f. A(x) satisfies A(x)^2 = 1 + 4*x*A(x)^5.
%C A245112 Radius of convergence of g.f. A(x) is r = (3/5)^(5/2) / 6 where A(r) = sqrt(5/3).
%H A245112 Gi-Sang Cheon, Sung-Tae Jin, and Louis W. Shapiro, <a href="https://doi.org/10.1016/j.laa.2015.03.015">A combinatorial equivalence relation for formal power series</a>, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
%F A245112 a(n) = 4^n * binomial((5*n - 1)/2, n) / (3*n + 1).
%F A245112 G.f. A(x) satisfies: A(x) = sqrt(1 + 4*x^2*A(x)^8) + 2*x*A(x)^4.
%F A245112 Self convolution yields A214553.
%F A245112 G.f. A(x) = 1/x * series reversion of x*sqrt(1 - 4*x*C(4*x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the generating function of the Catalan numbers A000108. See A024491. - _Peter Bala_, Mar 27 2023
%F A245112 D-finite with recurrence 3*n*(n-1)*(3*n-1)*(3*n+1)*a(n) - 20*(5*n-9)*(5*n-3)*(5*n-7)*(5*n-1)*a(n-2) = 0. - _R. J. Mathar_, Nov 22 2024
%F A245112 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^8). - _Seiichi Manyama_, Jun 20 2025
%F A245112 a(n) ~ 2^(n-1/2) * 5^(5*n/2) / (3^(3*n/2+1) * n^(3/2) * sqrt(Pi)). - _Amiram Eldar_, Sep 02 2025
%e A245112 G.f.: A(x) =  = 1 + 2*x + 18*x^2 + 224*x^3 + 3230*x^4 + 50688*x^5 +...
%e A245112 where A(x)^2 = 1 + 4*x*A(x)^5:
%e A245112 A(x)^2 = 1 + 4*x + 40*x^2 + 520*x^3 + 7680*x^4 + 122360*x^5 +...
%e A245112 A(x)^5 = 1 + 10*x + 130*x^2 + 1920*x^3 + 30590*x^4 + 512512*x^5 +...
%e A245112 Related series:
%e A245112 A(x)^4 = 1 + 8*x + 96*x^2 + 1360*x^3 + 21120*x^4 + 347760*x^5 +...
%e A245112 A(x)^8 = 1 + 16*x + 256*x^2 + 4256*x^3 + 73216*x^4 + 1294560*x^5 +...
%e A245112 where A(x) = sqrt(1 + 4*x^2*A(x)^8) + 2*x*A(x)^4.
%t A245112 a[n_] := 4^n * Binomial[(5*n - 1)/2, n] / (3*n + 1); Array[a, 20, 0] (* _Amiram Eldar_, Sep 02 2025 *)
%o A245112 (PARI) /* From A(x)^2 = 1 + 4*x*A(x)^5 : */
%o A245112 {a(n) = local(A=1+x);for(i=1,n,A=sqrt(1 + 4*x*A^5 +x*O(x^n)));polcoeff(A,n)}
%o A245112 for(n=0,20,print1(a(n),", "))
%o A245112 (PARI) {a(n) = 4^n * binomial((5*n - 1)/2, n) / (3*n + 1)}
%o A245112 for(n=0,20,print1(a(n),", "))
%o A245112 (PARI) /* From A(x) = sqrt(1 + 4*x^2*A(x)^8) + 2*x*A(x)^4 : */
%o A245112 {a(n) = local(A=1+x);for(i=1,n,A = sqrt(1 + 4*x^2*A^8 +x*O(x^n)) + 2*x*A^4);polcoeff(A,n)}
%o A245112 for(n=0,20,print1(a(n),", "))
%Y A245112 Cf. A000108, A024491, A214553, A245113.
%K A245112 nonn,easy,changed
%O A245112 0,2
%A A245112 _Paul D. Hanna_, Jul 31 2014