OFFSET
1,2
COMMENTS
Conjecture: a(n) == 1 (mod 2) iff n = 2^k - 1 for k >= 1.
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))^2 ).
(2) A(x)^4 = A( x*A(x)^3/((1 - A(x))^2*(1 - A(x)^2)^2) ).
(3) A(x)^8 = A( x*A(x)^7/((1 - A(x))^2*(1 - A(x)^2)^2*(1 - A(x)^4)^2) ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1)/Product_{k=0..n-1} (1 - A(x)^(2^k))^2 ) for n > 0.
(5) A(x) = x / Product_{n>=0} (1 - A(x)^(2^n))^2.
(6) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n))^(2*(n+1)).
(7) A(x) = Series_Reversion( x*B(x) ), where B(x) = Product_{n>=0} (1 - x^(2^n))^2 is the g.f. of A106407.
The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 2*2^n * A(r)^(2^n)/(1 - A(r)^(2^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(2^n))^2, where r = 0.1226755376200489039634571399751738538057851136283925... and A(r) = 0.2554610761187220452590974770035518728586331311247120...
Given r and A(r) above, A(r) also satisfies 1 = Sum_{n>=0} 2*(n+1)*2^n * A(r)^(2^n)/(1 + A(r)^(2^n)) ).
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 9*x^3 + 46*x^4 + 266*x^5 + 1636*x^6 + 10529*x^7 + 69974*x^8 + 476598*x^9 + 3309212*x^10 + ...
where A( x*A(x)/(1 - A(x))^2 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 128*x^5 + 797*x^6 + 5164*x^7 + 34506*x^8 + 235984*x^9 + 1643882*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))^2 = x - 2*x^2 - x^3 + 4*x^4 - 3*x^5 + 2*x^6 + 3*x^7 - 8*x^8 + x^9 + 6*x^10 + ... + A106407(n-1) * x^n + ...
thus,
x = A(x) * (1 - A(x))^2 * (1 - A(x)^2)^2 * (1 - A(x)^4)^2 * (1 - A(x)^8)^2 * (1 - A(x)^16)^2 * ... * (1 - A(x)^(2^n))^2 * ...
Also, notice that the square root of A(x)/x is an integral series
(A(x)/x)^(1/2) = 1 + x + 4*x^2 + 19*x^3 + 106*x^4 + 636*x^5 + 4024*x^6 + 26405*x^7 + 178096*x^8 + 1227018*x^9 + 8598424*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/5 at t = (1/5) * Product_{n>=0} (1 - 1/5^(2^n))^2 = 0.1175870125805304806733576532618445158357121658...
A(t) = 1/6 at t = (1/6) * Product_{n>=0} (1 - 1/6^(2^n))^2 = 0.1092311136132535692899568885022954464596243049...
A(1/9) = 0.17288740832245782814001741323630181133096513764543378...
A(1/10) = 0.1413215396171684711943139566840401836123301177323661...
A(1/11) = 0.1213541857717280074895334383318404648498876032468172...
PROG
(PARI) {a(n) = my(A = serreverse(x*prod(k=0, #binary(n), (1 - x^(2^k) + x*O(x^n))^2))); 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))^2 ) - 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