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A078534 Coefficients of power series that satisfies A(x)^5 - 25*x*A(x)^6 = 1, A(0)=1.

%I A078534 #28 Jun 21 2025 11:50:39
%S A078534 1,5,100,2625,78125,2502500,84150000,2929265625,104646953125,
%T A078534 3814697265625,141323284375000,5305403695312500,201382633183593750,
%U A078534 7715985752343750000,298023223876953125000,11591412585295166015625,453601640704152832031250
%N A078534 Coefficients of power series that satisfies A(x)^5 - 25*x*A(x)^6 = 1, A(0)=1.
%C A078534 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^(2*n-3) and a(2*n-1) = n^(4*n-2) (conjecture).
%C A078534 From _Emeric Deutsch_, Dec 10 2002: (Start)
%C A078534 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^(2*k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2*n-3) and a(2*n-1) = n^(4*n-2).
%C A078534 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 + x*A(x)^2. (End)
%H A078534 G. C. Greubel, <a href="/A078534/b078534.txt">Table of n, a(n) for n = 0..610</a>
%F A078534 a(n) = 5^(2*n)*binomial(6*n/5 - 4/5, n)/(n+1). - _Emeric Deutsch_, Dec 10 2002
%F A078534 a(n) ~ sqrt(3) * 6^(6*n/5 - 4/5) * 5^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 03 2014
%F A078534 From _Seiichi Manyama_, Jun 21 2025: (Start)
%F A078534 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7).
%F A078534 G.f.: ( (1/x) * Series_Reversion(x/(1+25*x)^(6/5)) )^(1/6). (End)
%e A078534 A(x)^5 - 25*x*A(x)^6 = 1 since A(x)^5 = 1 + 25*x + 750*x^2 + 24375*x^3 + 831250*x^4 + ... and A(x)^6 = 1 + 30*x + 975*x^2 + 33250*x^3 + ... also a(4) = 5^7, a(9) = 5^18 = 3814697265625.
%t A078534 Table[5^(2n) Binomial[(6n-4)/5,n]/(n+1),{n,0,25}]  (* _Harvey P. Dale_, Mar 27 2011 *)
%o A078534 (PARI) for(n=0,50, print1(5^(2*n)*binomial((6*n-4)/5, n)/(n+1), ", ")) \\ _G. C. Greubel_, Jan 30 2017
%Y A078534 Cf. A078531, A078532, A078533, A078535.
%K A078534 nonn
%O A078534 0,2
%A A078534 _Paul D. Hanna_, Nov 28 2002
%E A078534 More terms from _Harvey P. Dale_, Mar 27 2011