This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A212071 #23 Nov 22 2024 09:01:59
%S A212071 1,2,13,114,1150,12586,145299,1741844,21475146,270570300,3468352701,
%T A212071 45089941936,593082894768,7878407177270,105542811922950,
%U A212071 1424267372100456,19343105144742098,264182048662182420,3626176386241346070,49995713597946235350,692084935397470961346
%N A212071 G.f. satisfies: A(x) = (1 + x*A(x)^3)^2.
%C A212071 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
%H A212071 Michael De Vlieger, <a href="/A212071/b212071.txt">Table of n, a(n) for n = 0..856</a>
%H A212071 Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, <a href="https://doi.org/10.1016/j.laa.2015.03.015">A combinatorial equivalence relation for formal power series</a>, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
%H A212071 Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%H A212071 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>
%F A212071 a(n) = 2*binomial(6*n+2,n)/(6*n+2).
%F A212071 G.f.: A(x) = G(x)^2 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
%F A212071 a(n) = 2*binomial(6n+1, n-1)/n for n>0, a(0)=1. [_Bruno Berselli_, Jan 19 2014]
%F A212071 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
%F A212071 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
%e A212071 G.f.: A(x) = 1 + 2*x + 13*x^2 + 114*x^3 + 1150*x^4 + 12586*x^5 +...
%e A212071 Related expansions:
%e A212071 A(x)^3 = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...
%e A212071 A(x)^(1/2) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...
%t A212071 Table[c=6n+2;(2*Binomial[c,n])/c,{n,0,20}] (* _Harvey P. Dale_, Oct 14 2013 *)
%o A212071 (PARI) {a(n)=binomial(6*n+2,n) * 2/(6*n+2)}
%o A212071 for(n=0, 40, print1(a(n), ", "))
%o A212071 (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 A212071 Cf. A002295, A212072, A212073, A130564.
%K A212071 nonn,easy
%O A212071 0,2
%A A212071 _Paul D. Hanna_, Apr 29 2012