cp's OEIS Frontend

A212073 G.f. satisfies: A(x) = (1 + x*A(x)^(3/2))^4.

%I A212073 #12 Nov 22 2024 09:20:44
%S A212073 1,4,30,280,2925,32736,383838,4654320,57887550,734405100,9467075926,
%T A212073 123648163392,1632743088275,21761329287600,292362576381900,
%U A212073 3955219615609056,53834425161872586,736687428853685400,10129401435828605700,139876690363085200200
%N A212073 G.f. satisfies: A(x) = (1 + x*A(x)^(3/2))^4.
%C A212073 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 A212073 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>
%F A212073 a(n) = 4*binomial(6*n+4,n)/(6*n+4).
%F A212073 G.f. A(x) = G(x)^4 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
%F A212073 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 A212073 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 A212073 G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...
%e A212073 Related expansions:
%e A212073 A(x)^(3/2) = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...
%e A212073 A(x)^(1/4) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...
%t A212073 m = 20; A[_] = 0;
%t A212073 Do[A[x_] = (1 + x*A[x]^(3/2))^4 + O[x]^m, {m}];
%t A212073 CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 20 2019 *)
%o A212073 (PARI) {a(n)=binomial(6*n+4,n) * 4/(6*n+4)}
%o A212073 for(n=0, 40, print1(a(n), ", "))
%o A212073 (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 A212073 Cf. A002295, A212071, A212072, A130564, A069271, A118970, A233834, A234465, A234510, A234571, A235339.
%K A212073 nonn,easy
%O A212073 0,2
%A A212073 _Paul D. Hanna_, Apr 29 2012