A212073 - OEIS

%I #12 Nov 22 2024 09:20:44

%S 1,4,30,280,2925,32736,383838,4654320,57887550,734405100,9467075926,

%T 123648163392,1632743088275,21761329287600,292362576381900,

%U 3955219615609056,53834425161872586,736687428853685400,10129401435828605700,139876690363085200200

%N G.f. satisfies: A(x) = (1 + x*A(x)^(3/2))^4.

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(p*n + r, n)/(p*n + r); this is the case p = 6, r = 4. The o.g.f. B(x) of the Fuss_catalan sequence a(n,p,r) satisfies B(x) = {1 + x*B(x)^(p/r)}^r. - _Peter Bala_, Oct 14 2015

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

%F a(n) = 4*binomial(6*n+4,n)/(6*n+4).

%F G.f. A(x) = G(x)^4 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

%F O.g.f. A(x) = 1/x * series reversion (x/C(x)^4), where C(x) is the o.g.f. for the Catalan numbers A000108. - _Peter Bala_, Oct 14 2015

%F D-finite with recurrence 5*n*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n) -72*(6*n-1)*(3*n-1)*(2*n+1)*(3*n+1)*(6*n+1)*a(n-1)=0. - _R. J. Mathar_, Nov 22 2024

%e G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...

%e Related expansions:

%e A(x)^(3/2) = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...

%e A(x)^(1/4) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...

%t m = 20; A[_] = 0;

%t Do[A[x_] = (1 + x*A[x]^(3/2))^4 + O[x]^m, {m}];

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

%o (PARI) {a(n)=binomial(6*n+4,n) * 4/(6*n+4)}

%o for(n=0, 40, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=1+4*x); for(i=1, n, A=(1+x*A^(3/2))^4+x*O(x^n)); polcoeff(A, n)}

%Y Cf. A002295, A212071, A212072, A130564, A069271, A118970, A233834, A234465, A234510, A234571, A235339.

%K nonn,easy,changed

%O 0,2

%A _Paul D. Hanna_, Apr 29 2012