A361762 Expansion of g.f. A(x) satisfying A(x)^3 = A( x^3/(1 - 3*x)^3 ) / (1 - 3*x).

1, 1, 2, 5, 15, 52, 197, 779, 3135, 12709, 51757, 211761, 871022, 3603282, 14992067, 62719588, 263724900, 1114107925, 4726879206, 20135644606, 86099626270, 369492052236, 1591170063412, 6875211016868, 29803706856996, 129607445296468, 565362988510604, 2473576310166981

Offset: 0

Author

Paul D. Hanna, Mar 23 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 - 3*r)^(3/2) with A(r) = 1/r^(1/3) = 1.6716998816571609697481497812195572... so that A(r)^3 = A(r)/(1 - 3*r) and r = (52 - (324*sqrt(717) + 8108)^(1/3) + (324*sqrt(717) - 8108)^(1/3))/162 = 0.214054846272632706742187569443388024...

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 197*x^6 + 779*x^7 + 3135*x^8 + 12709*x^9 + 51757*x^10 + 211761*x^11 + 871022*x^12 + ...
such that A(x)^3 = A( x^3/(1 - 3*x)^3 ) / (1 - 3*x).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 93*x^4 + 333*x^5 + 1271*x^6 + 5064*x^7 + 20673*x^8 + 85460*x^9 + ... + A361763(n+1)*x^n + ...
A( x^3/(1 - 3*x)^3 ) = 1 + x^3 + 9*x^4 + 54*x^5 + 272*x^6 + 1251*x^7 + 5481*x^8 + 23441*x^9 + 99279*x^10 + ...
SPECIFIC VALUES.
A(1/5) = ( 5/2 * A(1/8) )^(1/3) = 1.431256341682946446458148822310720...
A(1/5) = (1 - 3/5)^(-1/3) * (1 - 3/8)^(-1/9) * (1 - 3/125)^(-1/27) * (1 - 3/1815848)^(-1/81) * ...
A(1/6) = ( 2 * A(1/27) )^(1/3) = 1.2765282682430983587479124671832773...
A(1/6) = (1 - 3/6)^(-1/3) * (1 - 3/27)^(-1/9) * (1 - 3/13824)^(-1/27) * (1 - 3/2640087986661)^(-1/81) * ...
A(1/9) = ( 3/2 * A(1/216) )^(1/3) = 1.146494555403917024085906029391966218...
A(1/12) = ( 4/3 * A(1/729) )^(1/3) = 1.101146836396635655557234214350215617...
		

Links

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

Crossrefs

Cf. A361763, A361764, A000108.

Programs

PARI

{a(n) = my(A=1); for(i=1, #binary(n+1), A = ( subst(A, x, x^3/(1 - 3*x +x*O(x^n))^3 )/(1 - 3*x +x*O(x^n)) )^(1/3) ); polcoeff(H=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)^3 = A( x^3/(1 - 3*x)^3 ) / (1 - 3*x).

(2) A(x^3) = A( x/(1 + 3*x) )^3 / (1 + 3*x).

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