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

1, 5, 25, 125, 625, 3130, 15800, 81625, 443125, 2609375, 16984500, 121023875, 914745625, 7093331250, 55129765625, 424092582500, 3212747690625, 23952422065625, 176059004593750, 1279867522656250, 9237023201350000, 66454031585359375, 478427499949687500, 3458191615224687500

Offset: 1

Author

Paul D. Hanna, Mar 27 2024

Keywords

• nonn

Examples

			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 + ...
where A(x)^5 = A( x*A(x)^4 + 5*x*A(x)^5 ).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A356782, A370439, A370545.

Programs

PARI

{a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);
F=Ser(A); A[#A] = polcoeff(subst(F,x,x*F^4 + 5*x*F^5) - F^5,#A+3) );A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1.a) A(x)^5 = A( x*A(x)^4 * (1 + 5*A(x)) ).

(1.b) A(x)^25 = A( x*A(x)^24 * (1 + 5*A(x))*(1 + 5*A(x)^5) ).

(1.c) A(x)^125 = A( x*A(x)^124 * (1 + 5*A(x))*(1 + 5*A(x)^5)*(1 + 5*A(x)^25) ).

(1.d) A(x)^(5^n) = A( x*A(x)^(5^n-1) * Product_{k=0..n-1} (1 + 5*A(x)^(5^k)) ).

(2) A(x) = x * Product_{n>=0} (1 + 5*A(x)^(5^n)).

(3) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 5*x^(5^n)) ).