A361764 Expansion of g.f. A(x) satisfying A(x)^5 = A( x^5/(1 - 5*x)^5 ) / (1 - 5*x).

1, 1, 3, 11, 44, 185, 806, 3627, 16926, 82615, 425633, 2325804, 13438568, 81258283, 507109592, 3223435416, 20655599675, 132496854084, 847152571284, 5386490329194, 34026141582719, 213512516149309, 1331393810596499, 8255968489237781, 50955585198416275, 313329163267012645

Offset: 0

Author

Paul D. Hanna, Mar 24 2023

Keywords

• nonn

Comments

Related Catalan identity: C(x)^2 = C( x^2/(1 - 2*x)^2 ) / (1 - 2*x), where 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/r^(1/5) = 1.451902871451714... so that A(r)^5 = A(r)/(1 - 5*r) and r = 0.1549930338264677513709380922535...

Examples

			G.f.: A(x) = 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 + ...
such that A(x)^5 = A( x^5/(1 - 5*x)^5 ) / (1 - 5*x).
RELATED SERIES.
A(x)^5 = 1 + 5*x + 25*x^2 + 125*x^3 + 625*x^4 + 3126*x^5 + 15655*x^6 + 78650*x^7 + 397625*x^8 + 2031875*x^9 + 10553128*x^10 + ...
A( x^5/(1 - 5*x)^5 ) = 1 + x^5 + 25*x^6 + 375*x^7 + 4375*x^8 + 43750*x^9 + 393753*x^10 + 3281400*x^11 + 25785375*x^12 + ...
SPECIFIC VALUES.
A(1/7) = ( 7/2 * A(1/32) )^(1/5) = 1.293495906485927953020670787280...
A(1/7) = (1 - 5/7)^(-1/5) * (1 - 5/32)^(-1/25) * (1 - 5/14348907)^(-1/125) * (1 - 5/14348902^5)^(-1/625) * ...
A(1/8) = ( 8/3 * A(1/243) )^(1/5) = 1.21774097368643014934892826038499995...
A(1/8) = (1 - 5/8)^(-1/5) * (1 - 5/243)^(-1/25) * (1 - 5/763633171168)^(-1/125) * (1 - 5/763633171163^5)^(-1/625) * ...
A(1/10) = ( 2 * A(1/3125) )^(1/5) = 1.14877193292427434012390599513357372...
A(1/10) = (1 - 5/10)^(-1/5) * (1 - 5/3125)^(-1/25) * (1 - 5/295646655283200000)^(-1/125) * (1 - 5/295646655283199995^5)^(-1/625) * ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A361765, A361762, A091200, A000108.

Programs

PARI

{a(n) = my(A=1); for(i=1, #binary(n+1), A = ( subst(A, x, x^5/(1 - 5*x +x*O(x^n))^5 )/(1 - 5*x +x*O(x^n)) )^(1/5) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

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

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

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