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 A078532 #33 Jul 30 2025 05:05:13
%S A078532 1,3,27,315,4158,59049,880308,13586859,215233605,3479417370,
%T A078532 57168561996,951892141473,16026585711660,272383068872700,
%U A078532 4666865660812044,80521573261807755,1397858693681272230,24398716826612190447,427921056863230599900,7537621933880388620010
%N A078532 Coefficients of power series that satisfies A(x)^3 - 9*x*A(x)^4 = 1, A(0)=1.
%C A078532 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 A078532 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 A078532 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 A078532 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 A078532 Self-convolution cube yields A214668.
%H A078532 G. C. Greubel, <a href="/A078532/b078532.txt">Table of n, a(n) for n = 0..750</a>
%F A078532 a(n) = 3^(2n)*binomial(4n/3-2/3, n)/(n+1). - _Emeric Deutsch_, Dec 10 2002
%F A078532 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 A078532 a(n) ~ 2^(8*n/3-5/6) * 3^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 03 2014
%F A078532 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
%F A078532 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^5). - _Seiichi Manyama_, Jun 20 2025
%e A078532 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 A078532 Table[3^(2n) Binomial[(4n-2)/3,n]/(n+1),{n,0,20}] (* _Harvey P. Dale_, Nov 03 2011 *)
%o A078532 (PARI) for(n=0,25, print1(9^n * binomial((4*n-2)/3, n)/(n+1), ", ")) \\ _G. C. Greubel_, Jan 26 2017
%Y A078532 Cf. A214668, A078531, A078533, A078534, A078535.
%Y A078532 Cf. A004987, A008931, A245114, A247029, A376282, A376636.
%Y A078532 Cf. A004990.
%K A078532 nonn,easy
%O A078532 0,2
%A A078532 _Paul D. Hanna_, Nov 28 2002
%E A078532 More terms from _Harvey P. Dale_, Nov 03 2011