OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..520
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x*A(x)/(1 - A(x))^3 ).
(2) A(x)^4 = A( x*A(x)^3/((1 - A(x))^3*(1 - A(x)^2)^3) ).
(3) A(x)^8 = A( x*A(x)^7/((1 - A(x))^3*(1 - A(x)^2)^3*(1 - A(x)^4)^3) ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1)/Product_{k=0..n-1} (1 - A(x)^(2^k))^3 ) for n > 0.
(5) A(x) = x / Product_{n>=0} (1 - A(x)^(2^n))^3.
(6) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n))^(3*(n+1)).
(7) A(x) = Series_Reversion( x*B(x) ), where B(x) = Product_{n>=0} (1 - x^(2^n))^3 is the g.f. of A373308.
The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 3*2^n * A(r)^(2^n)/(1 - A(r)^(2^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(2^n))^3, where r = 0.090173114826637655491436994778921911119292413640909... and A(r) = 0.197474208053634831172176658351098789075712647862486...
Given r and A(r) above, A(r) also satisfies 1 = Sum_{n>=0} 3*(n+1)*2^n * A(r)^(2^n)/(1 + A(r)^(2^n)) ).
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 18*x^3 + 127*x^4 + 999*x^5 + 8376*x^6 + 73400*x^7 + 664143*x^8 + 6157467*x^9 + 58190531*x^10 + ...
where A( x*A(x)/(1 - A(x))^3 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 45*x^4 + 362*x^5 + 3084*x^6 + 27318*x^7 + 249149*x^8 + 2323968*x^9 + 22067697*x^10 + ...
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^(2^n))^3 = x - 3*x^2 + 8*x^4 - 9*x^5 + 3*x^6 + 8*x^7 - 24*x^8 + 15*x^9 + 19*x^10 + ... + A373308(n-1) * x^n + ...
thus,
x = A(x) * (1 - A(x))^3 * (1 - A(x)^2)^3 * (1 - A(x)^4)^3 * (1 - A(x)^8)^3 * (1 - A(x)^16)^3 * ... * (1 - A(x)^(2^n))^3 * ...
Also, notice that the cube root of A(x)/x is an integral series
(A(x)/x)^(1/3) = 1 + x + 5*x^2 + 32*x^3 + 239*x^4 + 1937*x^5 + 16578*x^6 + 147408*x^7 + 1348465*x^8 + 12608851*x^9 + 119972595*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = (1/6) * Product_{n>=0} (1 - 1/6^(2^n))^3 = 0.0884290923082561345726735004152032422138677544...
A(t) = 1/7 at t = (1/7) * Product_{n>=0} (1 - 1/7^(2^n))^3 = 0.0844605844040460521136280418653467784637497846...
A(t) = 1/8 at t = (1/8) * Product_{n>=0} (1 - 1/8^(2^n))^3 = 0.0798174217593180496284155971364088109289815675...
A(1/12) = 0.1379538716718371951653031812720929490038524971492263...
A(1/13) = 0.1160657279647048938238673646663527089747582497393475...
A(1/14) = 0.1016889922856297159061963243507242491941351481713051...
PROG
(PARI) {a(n) = my(A = serreverse(x*prod(k=0, #binary(n), (1 - x^(2^k) + x*O(x^n))^3))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( subst(Ser(A), x, x*Ser(A)/(1 - Ser(A))^3 ) - Ser(A)^2, #A)); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 25 2024
STATUS
approved