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

1, 4, 30, 280, 2925, 32736, 383838, 4654320, 57887550, 734405100, 9467075926, 123648163392, 1632743088275, 21761329287600, 292362576381900, 3955219615609056, 53834425161872586, 736687428853685400, 10129401435828605700, 139876690363085200200

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..19.

Wikipedia, Fuss-Catalan number

Formula

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

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

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

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

Example

G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...
Related expansions:
A(x)^(3/2) = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...
A(x)^(1/4) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...

Mathematica

m = 20; A[_] = 0;
Do[A[x_] = (1 + x*A[x]^(3/2))^4 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 20 2019 *)

Prog

(PARI) {a(n)=binomial(6*n+4, n) * 4/(6*n+4)}
for(n=0, 40, print1(a(n), ", "))
(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)}

Crossrefs

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

Sequence in context: A370346 A375172 A274665 * A172392 A127130 A052631

Adjacent sequences: A212070 A212071 A212072 * A212074 A212075 A212076

Keyword

nonn,easy,changed

Author

Paul D. Hanna, Apr 29 2012

Status

approved