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 A078535 #23 Jun 21 2025 11:50:28
%S A078535 1,6,162,5760,232254,10077696,458960580,21634449408,1046465787510,
%T A078535 51644846702592,2590092194793948,131621703842267136,
%U A078535 6762649550214036780,350714987252652441600,18334388441036020419720,965148007553698721955840,51116742846877582931249574
%N A078535 Coefficients of power series that satisfies A(x)^6 - 36x*A(x)^7 = 1, A(0)=1.
%C A078535 If A(x)=sum_{k=1..inf} 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 A078535 If A(x)=sum_{k=1..inf} 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 A078535 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
%H A078535 Andrew Howroyd, <a href="/A078535/b078535.txt">Table of n, a(n) for n = 0..200</a>
%F A078535 a(n) = 6^(2n)*binomial(7n/6-5/6, n)/(n+1). - _Emeric Deutsch_, Dec 10 2002
%F A078535 a(n) ~ 7^(7*n/6-1/3) * 6^n / (sqrt(2*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 03 2014
%F A078535 From _Seiichi Manyama_, Jun 21 2025: (Start)
%F A078535 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^8).
%F A078535 G.f.: ( (1/x) * Series_Reversion(x/(1+36*x)^(7/6)) )^(1/7). (End)
%e A078535 A(x)^6 - 36x*A(x)^7 = 1 since A(x)^6 = 1 +36x +1512x^2 +68040x^3 +3193344x^4 +... and A(x)^7 = 1 +42x +1890x^2 +88704x^3 +... also a(5)=6^9, a(11)=6^22 = 131621703842267136.
%t A078535 Table[6^(2*n)*Binomial[7*n/6-5/6, n]/(n+1),{n,0,20}] (* _Vaclav Kotesovec_, Dec 03 2014 *)
%o A078535 (PARI) a(n) = {6^(2*n)*binomial((7*n-5)/6, n)/(n+1)} \\ _Andrew Howroyd_, Nov 05 2019
%Y A078535 Cf. A078531, A078532, A078533, A078534.
%K A078535 nonn
%O A078535 0,2
%A A078535 _Paul D. Hanna_, Nov 28 2002
%E A078535 Terms a(13) and beyond from _Andrew Howroyd_, Nov 05 2019