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A370546 Expansion of g.f. satisfying A(x) = A( x*A(x)^4 + 5*x*A(x)^5 )^(1/5).

%I A370546 #8 Mar 27 2024 21:25:08
%S A370546 1,5,25,125,625,3130,15800,81625,443125,2609375,16984500,121023875,
%T A370546 914745625,7093331250,55129765625,424092582500,3212747690625,
%U A370546 23952422065625,176059004593750,1279867522656250,9237023201350000,66454031585359375,478427499949687500,3458191615224687500
%N A370546 Expansion of g.f. satisfying A(x) = A( x*A(x)^4 + 5*x*A(x)^5 )^(1/5).
%H A370546 Paul D. Hanna, <a href="/A370546/b370546.txt">Table of n, a(n) for n = 1..400</a>
%F A370546 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370546 (1.a) A(x)^5 = A( x*A(x)^4 * (1 + 5*A(x)) ).
%F A370546 (1.b) A(x)^25 = A( x*A(x)^24 * (1 + 5*A(x))*(1 + 5*A(x)^5) ).
%F A370546 (1.c) A(x)^125 = A( x*A(x)^124 * (1 + 5*A(x))*(1 + 5*A(x)^5)*(1 + 5*A(x)^25) ).
%F A370546 (1.d) A(x)^(5^n) = A( x*A(x)^(5^n-1) * Product_{k=0..n-1} (1 + 5*A(x)^(5^k)) ).
%F A370546 (2) A(x) = x * Product_{n>=0} (1 + 5*A(x)^(5^n)).
%F A370546 (3) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 5*x^(5^n)) ).
%e A370546 G.f.: A(x) = x + 5*x^2 + 25*x^3 + 125*x^4 + 625*x^5 + 3130*x^6 + 15800*x^7 + 81625*x^8 + 443125*x^9 + 2609375*x^10 + 16984500*x^11 + 121023875*x^12 + ...
%e A370546 where A(x)^5 = A( x*A(x)^4 + 5*x*A(x)^5 ).
%o A370546 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);
%o A370546 F=Ser(A); A[#A] = polcoeff(subst(F,x,x*F^4 + 5*x*F^5) - F^5,#A+3) );A[n+1]}
%o A370546 for(n=1,30, print1(a(n),", "))
%Y A370546 Cf. A356782, A370439, A370545.
%K A370546 nonn
%O A370546 1,2
%A A370546 _Paul D. Hanna_, Mar 27 2024