A372531 Expansion of g.f. A(x) satisfying A(x)^3 = A( x*A(x)^2/(1 - A(x)) ).

1, 1, 2, 6, 19, 63, 220, 795, 2942, 11100, 42547, 165204, 648423, 2568522, 10255044, 41226054, 166732446, 677922831, 2769487183, 11362238976, 46794199487, 193387049685, 801742251778, 3333468469185, 13896609698686, 58073938493679, 243238872937589, 1020921149848044

Offset: 1

Author

Paul D. Hanna, May 14 2024

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 63*x^6 + 220*x^7 + 795*x^8 + 2942*x^9 + 11100*x^10 + 42547*x^11 + 165204*x^12 + ...
where A( x*A(x)^2/(1 - A(x)) ) = A(x)^3.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 111*x^7 + 405*x^8 + 1511*x^9 + 5742*x^10 + 22131*x^11 + 86310*x^12 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(3^n)) = x - x^2 - x^4 + x^5 - x^10 + x^11 + x^13 - x^14 - x^28 + x^29 + x^31 - x^32 + x^37 - x^38 - x^40 + x^41 - x^82 + ... + (-1)^A010060(n-1)*x^(A005836(n) + 1) + ...
thus,
x = A(x) * (1 - A(x)) * (1 - A(x)^3) * (1 - A(x)^9) * (1 - A(x)^27) * (1 - A(x)^81) * ... * (1 - A(x)^(3^n)) * ...
SPECIFIC VALUES.
A(t) = 2/5 at t = (2/5) * Product_{n>=0} (1 - (2/5)^(3^n)) = 0.224581111967794306351236678951339491766788581...
A(t) = 1/3 at t = (1/3) * Product_{n>=0} (1 - 1/3^(3^n)) = 0.213980897639074346024397964153942364246900732...
A(t) = 1/4 at t = (1/4) * Product_{n>=0} (1 - 1/4^(3^n)) = 0.184569608420133580452570741795926229187208705...
A(1/5) = 0.2875735682779125398024437286065851185781152551441974155...
A(1/6) = 0.2142049226274852453913309509157678575015367219972692281...
A(1/7) = 0.1738274438474723142423128822604443845966959546404795042...
A(1/6)^3 = A(t) at t = (1/6)*A(1/6)^2/(1 - A(1/6)) = 0.0097319157371...
A(1/7)^3 = A(t) at t = (1/7)*A(1/7)^2/(1 - A(1/7)) = 0.0052247784954...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..730

Crossrefs

Cf. A372530, A010060, A005836.

Programs

PARI

/* From Series_Reversion( x * Product_{n>=0} (1 - x^(3^n)) ) */
{a(n) = my(A, M=ceil(log(n+1)/log(3))); A = serreverse( x * prod(m=0,M, 1 - x^(3^m)) + x*O(x^n) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

PARI

/* From A(x)^3 = A( x*A(x)^2/(1 - A(x)) ) */
{a(n) = my(A=[0, 1],F); for(i=1, n, A = concat(A, 0); F=Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^2/(1 - F) ) - F^3, #A+1) ); H=A; A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Formula

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

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

(2) A(x)^9 = A( x*A(x)^8/((1 - A(x))*(1 - A(x)^3)) ).

(3) A(x)^27 = A( x*A(x)^26/((1 - A(x))*(1 - A(x)^3)*(1 - A(x)^9)) ).

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

(5) A(x) = x / Product_{n>=0} (1 - A(x)^(3^n)).

(6) A(x) = Series_Reversion( x * Product_{n>=0} (1 - x^(3^n)) ).

(7) x = A(x) * Sum_{n>=1} (-1)^A010060(n-1) * A(x)^A005836(n), where A010060 is the Thue-Morse sequence and A005836 lists numbers whose base-3 representation contains no 2.

The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 3^n * A(r)^(3^n)/(1 - A(r)^(3^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(3^n)), where r = 0.225516184149820697566779292359148589008135334130979... and A(r) = 0.426459850024068213581384788221931983633117762268826...