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%I #22 Mar 24 2023 15:11:46
%S 1,3,27,315,4158,59049,880308,13586859,215233605,3479417370,
%T 57168561996,951892141473,16026585711660,272383068872700,
%U 4666865660812044,80521573261807755,1397858693681272230,24398716826612190447,427921056863230599900,7537621933880388620010
%N Coefficients of power series that satisfies A(x)^3 - 9*x*A(x)^4 = 1, A(0)=1.
%C If A(x) = Sum_{k>=1} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2) (conjecture).
%C If A(x) = Sum_{k>=1} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - _Emeric Deutsch_, Dec 10 2002
%C A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - _Emeric Deutsch_, Dec 10 2002
%C Radius of convergence of g.f. A(x) is r = 1/(3*4^(4/3)) where A(r) = 4^(1/3). - _Paul D. Hanna_, Jul 24 2012
%C Self-convolution cube yields A214668.
%C D-finite with recurrence n*(n-1)*(n+1)*a(n) -216*(4*n-5)*(2*n-1)*(4*n-11)*a(n-3)=0. - _R. J. Mathar_, Mar 24 2023
%H G. C. Greubel, <a href="/A078532/b078532.txt">Table of n, a(n) for n = 0..750</a>
%F a(n) = 3^(2n)*binomial(4n/3-2/3, n)/(n+1). - _Emeric Deutsch_, Dec 10 2002
%F Sequence with offset 1 is expansion of reversion of g.f. x*(1-9*x)^(1/3), which equals x times the g.f. of A004990.
%F a(n) ~ 2^(8*n/3-5/6) * 3^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 03 2014
%e A(x)^3 - 9x*A(x)^4 = 1 since A(x)^3 = 1 +9x +108x^2 +1458x^3 +21060x^4 +... and A(x)^4 = 1 +12x +162x^2 +2340x^3 +... also a(2)=3^3, a(5)=3^10.
%t Table[3^(2n) Binomial[(4n-2)/3,n]/(n+1),{n,0,20}] (* _Harvey P. Dale_, Nov 03 2011 *)
%o (PARI) for(n=0,25, print1(9^n * binomial(4*n-2)/3, n)/(n+1), ", ")) \\ _G. C. Greubel_, Jan 26 2017
%Y Cf. A214668, A078531, A078533, A078534, A078535.
%Y Cf. A004990.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 28 2002
%E More terms from _Harvey P. Dale_, Nov 03 2011
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