A214553 - OEIS

%I #26 Oct 20 2019 04:13:01

%S 1,4,40,520,7680,122360,2050048,35600400,635043840,11566760920,

%T 214221455360,4021962900592,76374500966400,1464312851075760,

%U 28307243610931200,551140224522544160,10797908842864705536,212721273248318069400,4211238736846158561280

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

%C Radius of convergence of g.f. A(x) is r = (3/5)^(5/2) / 6 where A(r) = 5/3.

%C 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 Robert Israel, <a href="/A214553/b214553.txt">Table of n, a(n) for n = 0..720</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>

%F a(n) = 4^n * binomial(5*n/2, n) / (3*n/2 + 1).

%F From _Peter Bala_, Oct 13 2015: (Start)

%F 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 Sqrt(A(x)) = 1/x * series reversion( x/sqrt(C(4*x)) ) is the o.g.f. for A245112. (End)

%e 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 Radius of convergence: r = (3/5)^(5/2)/6 = 0.046475800...

%e Related expansions:

%e A(x)^(5/2) = 1 + 10*x + 130*x^2 + 1920*x^3 + 30590*x^4 + 512512*x^5 +...

%e 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 seq(4^n*binomial(5*n/2,n)/(3*n/2+1),n=0..50); # _Robert Israel_, Oct 18 2015

%t m = 19; A[_] = 0;

%t Do[A[x_] = 1 + 4*x*A[x]^(5/2) + O[x]^m, {m}];

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 20 2019 *)

%o (PARI) {a(n)=4^n*binomial(5*n/2,n)/(3*n/2+1)}

%o (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 for(n=0,30,print1(a(n),", "))

%Y Cf. A000108, A214377, A245112.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Jul 20 2012