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A212071 G.f. satisfies: A(x) = (1 + x*A(x)^3)^2. 6

%I

%S 1,2,13,114,1150,12586,145299,1741844,21475146,270570300,3468352701,

%T 45089941936,593082894768,7878407177270,105542811922950,

%U 1424267372100456,19343105144742098,264182048662182420,3626176386241346070,49995713597946235350,692084935397470961346

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

%C 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

%D Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.

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

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

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

%F a(n) = 2*binomial(6n+1, n-1)/n for n>0, a(0)=1. [_Bruno Berselli_, Jan 19 2014]

%F 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

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

%e Related expansions:

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

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

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

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

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

%o (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)}

%Y Cf. A002295, A212072, A212073, A130564.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Apr 29 2012

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Last modified March 8 01:41 EST 2021. Contains 341934 sequences. (Running on oeis4.)