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A214553 G.f. A(x) satisfies A(x) = 1 + 4*x*A(x)^(5/2).

%I A214553 #38 Sep 02 2025 04:06:18
%S A214553 1,4,40,520,7680,122360,2050048,35600400,635043840,11566760920,
%T A214553 214221455360,4021962900592,76374500966400,1464312851075760,
%U A214553 28307243610931200,551140224522544160,10797908842864705536,212721273248318069400,4211238736846158561280
%N A214553 G.f. A(x) satisfies A(x) = 1 + 4*x*A(x)^(5/2).
%C A214553 Radius of convergence of g.f. A(x) is r = (3/5)^(5/2) / 6 where A(r) = 5/3.
%C A214553 The two-parameter Fuss-Catalan sequence is A_n(p,r) := r*binomial(n*p + r, n)/(n*p + r). This sequence is 4^n*A_n(5/2,1). - _Peter Bala_, Oct 15 2015
%H A214553 Robert Israel, <a href="/A214553/b214553.txt">Table of n, a(n) for n = 0..720</a>
%H A214553 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>.
%F A214553 a(n) = 4^n * binomial(5*n/2, n) / (3*n/2 + 1).
%F A214553 From _Peter Bala_, Oct 13 2015: (Start)
%F A214553 O.g.f. A(x) satisfies A(x) = C(4*x*sqrt(A)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.
%F A214553 Sqrt(A(x)) = 1/x * series reversion( x/sqrt(C(4*x)) ) is the o.g.f. for A245112. (End)
%F A214553 D-finite with recurrence 3*n*(n-1)*(3*n+2)*(3*n-2)*a(n) - 20*(5*n-4)*(5*n-8)*(5*n-2)*(5*n-6)*a(n-2) = 0. - _R. J. Mathar_, Nov 22 2024
%F A214553 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^4). - _Seiichi Manyama_, Jun 17 2025
%F A214553 a(n) ~ 2^(n+1/2) * 5^((5*n+1)/2) / (3^(3*(n+1)/2) * n^(3/2) * sqrt(Pi)). - _Amiram Eldar_, Sep 02 2025
%e A214553 G.f.: A(x) = 1 + 4*x + 40*x^2 + 520*x^3 + 7680*x^4 + 122360*x^5 + 2050048*x^6 +... where A(x) = 1 + 4*x*A(x)^(5/2).
%e A214553 Radius of convergence: r = (3/5)^(5/2)/6 = 0.046475800...
%e A214553 Related expansions:
%e A214553 A(x)^(5/2) = 1 + 10*x + 130*x^2 + 1920*x^3 + 30590*x^4 + 512512*x^5 +...
%e A214553 A(x)^(1/2) = 1 + 2*x + 18*x^2 + 224*x^3 + 3230*x^4 + 50688*x^5 + 840420*x^6 + 14483456*x^7 + 256856886*x^8 +...
%p A214553 seq(4^n*binomial(5*n/2,n)/(3*n/2+1),n=0..50); # _Robert Israel_, Oct 18 2015
%t A214553 m = 19; A[_] = 0;
%t A214553 Do[A[x_] = 1 + 4*x*A[x]^(5/2) + O[x]^m, {m}];
%t A214553 CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 20 2019 *)
%o A214553 (PARI) {a(n)=4^n*binomial(5*n/2,n)/(3*n/2+1)}
%o A214553 (PARI) {a(n)=local(A=1+x);for(i=1,n,A =1+4*x*(A+x*O(x^n))^(5/2));polcoeff(A,n)}
%o A214553 for(n=0,30,print1(a(n),", "))
%Y A214553 Cf. A000108, A135863, A214377, A245112.
%K A214553 nonn,easy,changed
%O A214553 0,2
%A A214553 _Paul D. Hanna_, Jul 20 2012