A212071 - OEIS

A212071

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

1, 2, 13, 114, 1150, 12586, 145299, 1741844, 21475146, 270570300, 3468352701, 45089941936, 593082894768, 7878407177270, 105542811922950, 1424267372100456, 19343105144742098, 264182048662182420, 3626176386241346070, 49995713597946235350, 692084935397470961346

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OFFSET

0,2

COMMENTS

The two parameter Fuss-Catalan sequence is A(n,p,r) := r*binomial(n*p + r, n)/(n*p + r), with o.g.f. G(p,r,x) = G(x) satisfying G(x) = {1 + x*G(x)^(p/r)}^r ; this is the case p = 6, r = 2. - Peter Bala, Oct 14 2015

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..856

Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.

Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

Wikipedia, Fuss-Catalan number

FORMULA

a(n) = 2*binomial(6*n+2,n)/(6*n+2).

G.f.: A(x) = G(x)^2 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

a(n) = 2*binomial(6n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]

A(x^2) = 1/x * series reversion (x/C(x^2)^2), where C(x) = (1 - sqrt( 1 - 4*x))/(2*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-2)*(5*n-1)*a(n) -72*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

EXAMPLE

G.f.: A(x) = 1 + 2*x + 13*x^2 + 114*x^3 + 1150*x^4 + 12586*x^5 +...

Related expansions:

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

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

MATHEMATICA

Table[c=6n+2; (2*Binomial[c, n])/c, {n, 0, 20}] (* Harvey P. Dale, Oct 14 2013 *)

PROG

(PARI) {a(n)=binomial(6*n+2, n) * 2/(6*n+2)}

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

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