OFFSET
1,2
COMMENTS
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).
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...
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x)^5 = A( x^5/(1 - 5*x)^5 ).
(2) A(x^5) = A( x/(1 + 5*x) )^5.
(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.
EXAMPLE
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 + ...
where
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 + ...
which equals A(x)^5.
RELATED SERIES.
Notice that the following fifth root is an integer series
( 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 + ...
SPECIFIC VALUES.
A(1/7) = A(1/32)^(1/5) = 0.5172818651818402813815396980...
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) * ...
A(1/8) = A(1/243)^(1/5) = 0.334722270350398633572525135166...
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) * ...
A(1/10) = A(1/3125)^(1/5) = 0.2000640615121819990127352003599...
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) * ...
PROG
(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)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 24 2023
STATUS
approved