A372530 - OEIS

%I #23 Jun 28 2024 05:06:09

%S 1,1,3,9,33,125,501,2065,8739,37685,165107,732681,3286679,14878885,

%T 67889851,311896993,1441536321,6698017445,31269529601,146601334841,

%U 689945263873,3258334336349,15436401872405,73341269533009,349381321611505,1668434132560765,7985390073708765

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

%C Compare to the following identities of the Catalan function C(x) = x + C(x)^2 (A000108):

%C (1) C(x)^2 = C( x*C(x)*(1 + C(x)) ),

%C (2) C(x)^4 = C( x*C(x)^3*(1 + C(x))*(1 + C(x)^2) ),

%C (3) C(x)^8 = C( x*C(x)^7*(1 + C(x))*(1 + C(x)^2)*(1 + C(x)^4) ),

%C (4) C(x)^(2^n) = C( x*C(x)^(2^n-1)*Product_{k=0..n-1} (1 + C(x)^(2^k)) ) for n > 0.

%H Paul D. Hanna, <a href="/A372530/b372530.txt">Table of n, a(n) for n = 1..520</a>

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

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

%F (2) A(x)^4 = A( x*A(x)^3/((1 - A(x))*(1 - A(x)^2)) ).

%F (3) A(x)^8 = A( x*A(x)^7/((1 - A(x))*(1 - A(x)^2)*(1 - A(x)^4)) ).

%F (4) A(x)^(2^n) = A( x*A(x)^(2^n-1)/Product_{k=0..n-1} (1 - A(x)^(2^k)) ) for n > 0.

%F (5) A(x) = x / Product_{n>=0} (1 - A(x)^(2^n)).

%F (6) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n))^(n+1). - _Paul D. Hanna_, Jun 26 2024

%F (7) A(x) = Series_Reversion( x * Product_{n>=0} (1 - x^(2^n)) ).

%F (8) x = Sum_{n>=1} (-1)^A010060(n-1) * A(x)^n, where A010060 is the Thue-Morse sequence.

%F The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 2^n * A(r)^(2^n)/(1 - A(r)^(2^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(2^n)), where r = 0.19736158352631556925015099049581233030702919287488... and A(r) = 0.37298513723316144189484491702105095014110332846051...

%F Given r and A(r) above, A(r) also satisfies 1 = Sum_{n>=0} (n+1)*2^n * A(r)^(2^n)/(1 + A(r)^(2^n)) ). - _Paul D. Hanna_, Jun 26 2024

%e G.f.: A(x) = x + x^2 + 3*x^3 + 9*x^4 + 33*x^5 + 125*x^6 + 501*x^7 + 2065*x^8 + 8739*x^9 + 37685*x^10 + 165107*x^11 + 732681*x^12 + ...

%e where A( x*A(x)/(1 - A(x)) ) = A(x)^2.

%e RELATED SERIES.

%e Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then

%e R(x) = x * Product_{n>=0} (1 - x^(2^n)) = x - x^2 - x^3 + x^4 - x^5 + x^6 + x^7 - x^8 - x^9 + x^10 + x^11 - x^12 + x^13 - x^14 - x^15 + x^16 + ... + (-1)^A010060(n-1) * x^n + ...

%e thus,

%e x = A(x) * (1 - A(x)) * (1 - A(x)^2) * (1 - A(x)^4) * (1 - A(x)^8) * (1 - A(x)^16) * ... * (1 - A(x)^(2^n)) * ...

%e SPECIFIC VALUES.

%e A(t) = 1/3 at t = (1/3) * Product_{n>=0} (1 - 1/3^(2^n)) = 0.195062471888103139123433255203480726664398592...

%e A(t) = 1/4 at t = (1/4) * Product_{n>=0} (1 - 1/4^(2^n)) = 0.175091932719784804433277263483089433821043251...

%e A(1/6) = 0.2285942310240955503097133963953487564542629539800372181...

%e A(1/7) = 0.1803372891149269875688065840927292319030238580575714990...

%e A(1/8) = 0.1506715662175837437127190414569072051853697889895576799...

%e A(1/6)^2 = A(t) at t = (1/6)*A(1/6)/(1 - A(1/6)) = 0.0493891023845...

%e A(1/7)^2 = A(t) at t = (1/7)*A(1/7)/(1 - A(1/7)) = 0.0314305744685...

%o (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0); F=Ser(A);

%o A[#A] = polcoeff( subst(F,x, x*F/(1 - F) ) - F^2, #A) ); H=A; A[n+1]}

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A373312, A373313, A372531, A371713, A371709, A010060.

%K nonn

%O 1,3

%A _Paul D. Hanna_, May 13 2024