A361765 - OEIS

%I #15 Apr 17 2023 22:15:08

%S 1,5,25,125,625,3126,15655,78650,397625,2031875,10553128,56047040,

%T 306020575,1723544750,10015548750,59871903136,366244516505,

%U 2278239803025,14324961668875,90586470006875,573925269278169,3633524853973370,22949197586894725,144473478898021750

%N Expansion of g.f. A(x) satisfying A(x)^5 = A( x^5/(1 - 5*x)^5 ).

%C Related Catalan identity: F(x)^2 = F( x^2/(1 - 2*x)^2 ), where F(x) = x*C(x)^2 and  C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

%C Radius of convergence of g.f. A(x) is r where r is the real root of r = (1 - 5*r)^(5/4) with A(r) = 1 and r = 0.1549930338264677513709380922535...

%H Paul D. Hanna, <a href="/A361765/b361765.txt">Table of n, a(n) for n = 1..500</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

%F (1) A(x)^5 = A( x^5/(1 - 5*x)^5 ).

%F (2) A(x^5) = A( x/(1 + 5*x) )^5.

%F (3) A(x) = x * Product_{n>=0} 1/(1 - 5/F(n,x))^(1/5^n), where F(0,x) = 1/x, F(m,x) = (F(m-1,x) - 5)^5 for m > 0.

%e G.f.: A(x) = x + 5*x^2 + 25*x^3 + 125*x^4 + 625*x^5 + 3126*x^6 + 15655*x^7 + 78650*x^8 + 397625*x^9 + 2031875*x^10 + 10553128*x^11 + ...

%e where

%e A( x^5/(1 - 5*x)^5 ) = x^5 + 25*x^6 + 375*x^7 + 4375*x^8 + 43750*x^9 + 393755*x^10 + 3281500*x^11 + 25788125*x^12 + 193496875*x^13 + ...

%e which equals A(x)^5.

%e RELATED SERIES.

%e Notice that the following fifth root is an integer series

%e ( A(x)/x )^(1/5) = 1 + x + 3*x^2 + 11*x^3 + 44*x^4 + 185*x^5 + 806*x^6 + 3627*x^7 + 16926*x^8 + 82615*x^9 + 425633*x^10 + ... + A361764(n)*x^n + ...

%e SPECIFIC VALUES.

%e A(1/7) = A(1/32)^(1/5) = 0.5172818651818402813815396980...

%e A(1/7) = (1/7) * (1 - 5/7)^(-1) * (1 - 5/32)^(-1/5) * (1 - 5/14348907)^(-1/25) * (1 - 5/14348902^5)^(-1/125) * ...

%e A(1/8) =  A(1/243)^(1/5) = 0.334722270350398633572525135166...

%e A(1/8) = (1/8) * (1 - 5/8)^(-1) * (1 - 5/243)^(-1/5) * (1 - 5/763633171168)^(-1/25) * (1 - 5/763633171163^5)^(-1/125) * ...

%e A(1/10) = A(1/3125)^(1/5) = 0.2000640615121819990127352003599...

%e A(1/10) = (1/10) * (1 - 5/10)^(-1) * (1 - 5/3125)^(-1/5) * (1 - 5/295646655283200000)^(-1/25) * (1 - 5/295646655283199995^5)^(-1/125) * ...

%o (PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = ( subst(A, x, x^5/(1 - 5*x +x*O(x^n))^5 ) )^(1/5) ); polcoeff(A, n)}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A361764, A352704, A361763.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 24 2023