%I A078535
%S A078535 1,6,162,5760,232254,10077696,458960580,21634449408,1046465787510,
%T A078535 51644846702592,2590092194793948,131621703842267136,
%U A078535 6762649550214036780
%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
%F A078535 a(n)=6^(2n)*binomial(7n/6-5/6, n)/(n+1) - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 10 2002
%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.
%Y A078535 Cf. A078531, A078532, A078533, A078534.
%Y A078535 Sequence in context: A120277 A015086 A052466 this_sequence A143534 A104729 
               A106661
%Y A078535 Adjacent sequences: A078532 A078533 A078534 this_sequence A078536 A078537 
               A078538
%K A078535 nonn
%O A078535 0,2
%A A078535 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 28 2002