A380554 G.f. A(x) satisfies A(x)^4 = A( A(x)^3 * x/(1-x) ).

1, 1, 1, 1, 2, 6, 16, 36, 75, 163, 391, 991, 2498, 6150, 15016, 37116, 93482, 238154, 608074, 1551370, 3964200, 10176384, 26261500, 68034484, 176661828, 459534596, 1197777556, 3129475636, 8195867902, 21508247446, 56540427826, 148863643466, 392539322259, 1036662269875, 2741706892035

Offset: 1

Author

Paul D. Hanna, Jan 26 2025

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 16*x^7 + 36*x^8 + 75*x^9 + 163*x^10 + 391*x^11 + 991*x^12 + 2498*x^13 + 6150*x^14 + 15016*x^15 + ...
where A(x)^4 = A( A(x)^3 * x/(1-x) );
also, A(x) = x*(1 + A(x) + A(x)^4 + A(x)^16 + A(x)^64 + ...).
RELATED SERIES.
x/(1 + x + x^4 + x^16 + x^64 + ...) = x - x^2 + x^3 - x^4 + x^6 - 2*x^7 + 3*x^8 - 3*x^9 + 2*x^10 - 3*x^12 + 6*x^13 - 8*x^14 + 8*x^15 - 5*x^16 + ...
where x = A( x/(1 + x + x^4 + x^16 + x^64 + ...) ).
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 42*x^8 + 112*x^9 + 288*x^10 + ...
A(x)^4 = x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 39*x^8 + 88*x^9 + 228*x^10 + 600*x^11 + ...
SPECIFIC VALUES.
A(t) = 7/10 at t = 0.36018915820185609929548309671397017657231396...
  where (7/10)^4 = A( (7/10)^3*t/(1-t) )
  and t = (7/10)/(1 + Sum_{n>=0} (7/10)^(4^n)).
A(t) = 2/3 at t = 0.357324077294579321123715825007257976292387856...
  where 16/81 = A( (8/27)*t/(1-t) )
  and t = (2/3)/(1 + Sum_{n>=0} (2/3)^(4^n)).
A(t) = 1/2 at t = 0.319996875030517280093584464262123092506355813...
  where 1/16 = A( (1/8)*t/(1-t) )
  and t = (1/2)/(1 + Sum_{n>=0} (1/2)^(4^n)).
A(t) = 1/3 at t = 0.247706417742171319902767393551872977585317906...
  where 1/81 = A( (1/27)*t/(1-t) )
  and t = (1/3)/(1 + Sum_{n>=0} (1/3)^(4^n)).
A(1/3) = 0.54373202136840396341881074287828877295481851718413...
  where A(1/3)^4 = A( A(1/3)^3*(1/2) ).
A(1/4) = 0.33766677567921691723942758840979376280294197783058...
  where A(1/4)^4 = A( A(1/4)^3*(1/3) ).
A(1/5) = 0.25099215755350299738032710744403195608988446686839...
A(1/6) = 0.20032206620931060989695576481191496886558371212657...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1030

Crossrefs

Cf. A075864, A374565.

Programs

PARI

/* A(x) = Series_Reversion( x/(1 + Sum_{n>=0} x^(4^n)) ) */
{a(n) = my(A = serreverse( x/(1 + sum(m=0,ceil(log(n+1)/log(4)), x^(4^m) +x*O(x^n))) )); polcoef(A,n)}
for(n=1, 40, print1(a(n), ", "))

PARI

/* A(x)^4 = A( A(x)^3 * x/(1-x) ) */
{a(n) = my(A=[1], Ax);
for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
A[#A] = -polcoeff( Ax^4 - subst(Ax, x, Ax^3*x/(1-x) ), #A+3) ); 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) = Series_Reversion( x/(1 + Sum_{n>=0} x^(4^n)) ).

(2) A(x) = x * (1 + Sum_{n>=0} A(x)^(4^n)).

(3) A(x) = x/(1-x) * (1 + Sum_{n>=1} A(x)^(4^n)).

(4) A(x)^4 = A( A(x)^3 * x/(1-x) ).

(5) A(x)^16 = A( A(x)^15 * x/(1 - x - x*A(x)^3) ).

(6) A(x)^64 = A( A(x)^63 * x/(1 - x - x*A(x)^3 - x*A(x)^15) ).

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

The radius of convergence r and A(r) satisfy r = 1/(Sum_{n>=0} 4^n*A(r)^(4^n-1)) and A(r) = A( A(r)^3*r/(1-r) )^(1/4), where r = 0.3613437470225014946622689597447779556234350427479140... and A(r) = 0.7371720020640001613320630406857895231048184830453856...