A374565 Expansion of g.f. A(x) satisfying A(x)^3 = A( x*A(x)^2/(1-x) ).

1, 1, 1, 2, 5, 11, 24, 57, 141, 350, 881, 2267, 5920, 15601, 41497, 111399, 301293, 819843, 2243058, 6167211, 17029473, 47200752, 131270283, 366195789, 1024380648, 2872770381, 8074967031, 22745832254, 64196912681, 181516532273, 514107418321, 1458407886019, 4143318012685

Offset: 1

Author

Paul D. Hanna, Jul 23 2024

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 24*x^7 + 57*x^8 + 141*x^9 + 350*x^10 + 881*x^11 + 2267*x^12 + 5920*x^13 + 15601*x^14 + 41497*x^15 + ...
where A(x)^3 = A( x*A(x)^2/(1-x) )
and A(x) = x + x*(A(x) + A(x)^3 + A(x)^9 + A(x)^27 + ... A(x)^(3^n) + ...).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 33*x^7 + 84*x^8 + 208*x^9 + 522*x^10 + 1341*x^11 + 3479*x^12 + 9078*x^13 + 23907*x^14 + 63560*x^15 + ...
Let B(x) be the series reversion of A(x), B(A(x)) = x, then
B(x) = x - x^2 + x^3 - 2*x^4 + 3*x^5 - 4*x^6 + 6*x^7 - 9*x^8 + 13*x^9 - 20*x^10 + ... + (-1)^(n-1)*A078932(n-1)*x^n + ...
where x/B(x) = 1 + x + x^3 + x^9 + x^27 + x^81 + ... + x^(3^n) + ...
F(x) = A(x/(1+x)) = x + x^4 + 3*x^7 + 13*x^10 + 67*x^13 + 378*x^16 + 2253*x^19 + 13947*x^22 + 88803*x^25 + 577903*x^28 + 3826870*x^31 + 25703868*x^34 + ...
where F(x)^3 = F( x*F(x)^2/(1 - x*F(x)^2) )
and F(x) = x + x*(F(x)^3 + F(x)^9 + F(x)^27 + ... + F(x)^(3^n) + ...).
SPECIFIC VALUES.
A(t) = 2/3 at t = 0.3351780091733165997365854281871805851976265481916...
where 8/27 = A( (4/9)*t/(1-t) )
and t = (2/3)/(1 + Sum_{n>=0} (2/3)^(3^n)).
A(t) = 1/2 at t = 0.3073229277642929985518391822746766756418592443672...
where 1/8 = A( (1/4)*t/(1-t) )
and t = (1/2)/(1 + Sum_{n>=0} (1/2)^(3^n)).
A(1/3) = 0.640317989282342396539425948311398871030928082061168...
where A(1/3)^3 = A( A(1/3)^2/2 ).
A(1/4) = 0.347324237093006237340030053166266719890703533474663...
where A(1/4)^3 = A( A(1/4)^2/3 ).
A(1/5) = 0.254102848699628177600720471035831153854183353627930...
where A(1/5)^3 = A( A(1/5)^2/4 ).
A(1/10) = 0.111264157881789221767410282888976753122883279205707...
where A(1/10)^3 = A( A(1/10)^2/9 ).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..801

Crossrefs

Cf. A078932, A075864.

Programs

PARI

{a(n) = my(A = serreverse(x/(1 + sum(n=0,ceil(log(n+1)/log(3)), x^(3^n)) + x^3*O(x^n)) )); polcoeff(A,n)}
for(n=1, 40, print1(a(n), ", "))

PARI

{a(n) = my(A=[1], Ax);
for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
A[#A] = -polcoeff( Ax^3 - subst(Ax, x, Ax^2*x/(1-x) ), #A+2) ); A[n]}
for(n=1, 40, print1(a(n), ", "))

Formula

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

(1) A(x) = x + x*Sum_{n>=0} A(x)^(3^n).

(2) A(x) = Series_Reversion( x/(1 + Sum_{n>=0} x^(3^n)) ).

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

(4) A(x)^9 = A( x*A(x)^8/(1 - x - x*A(x)^2) ).

(5) A(x)^27 = A( x*A(x)^26/(1 - x - x*A(x)^2 - x*A(x)^8) ).

(6) A(x)^(3^n) = A( x*A(x)^(3^n-1) / (1 - x*Sum_{k=0..n-1} A(x)^(3^k-1)) ) for n >= 1.

The radius of convergence r and A(r) satisfy r = 1/(Sum_{n>=0} 3^n*A(r)^(3^n-1)) and A(r) = A( A(r)^2*r/(1-r) )^(1/3), where r = 0.3359879296886914478616860912190963818298151003686099... and A(r) = 0.6985186992950193189255500784091315877737446624401085...