OFFSET
0,2
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1), which holds when 0 < |x| < 1.
Note that Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n is to be taken as the sum of two infinite series, P(x) + Q(x), where P(x) = Sum_{n=-oo..-1} n * x^n * (1 - x^n)^(n-1) * A(x)^n and Q(x) = Sum_{n=+1..+oo} n * x^n * (1 - x^n)^(n-1) * A(x)^n. The g.f. A(x) of this sequence satisfies the condition that P(x) + Q(x) = 0. The series Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n converges to zero when 0 < |x| < r where r < 1 is the radius of convergence of g.f. A(x). Upon reversing the sign of the index n, and so taking the same sum in reverse order from +oo to -oo, we obtain the equivalent series Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) / ((1 - x^n)^(n+1) * A(x)^n), the convergence of which is more clearly seen to hold when 0 < |x| < r < 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n, with a(0) = -1, satisfies the following relations.
(1) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n.
(2) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n/A(x))^(n-1).
(3) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (A(x) - x^n)^(n-1).
(4) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) / ( (1 - x^n)^(n+1) * A(x)^n ).
(5) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) * A(x)^n / (A(x) - x^n)^(n+1).
(6) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) * A(x)^n / (1 - x^n*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = -1 - 2*x - 4*x^2 - 8*x^3 - 8*x^4 - 6*x^5 + 40*x^6 + 132*x^7 + 400*x^8 + 504*x^9 + 76*x^10 - 4960*x^11 - 18528*x^12 - 56998*x^13 - 94176*x^14 - 58896*x^15 + 617216*x^16 + ...
such that
0 = ... - 3*(x*A(x))^(-3)/(1 - x^(-3))^4 - 2*(x*A(x))^(-2)/(1 - x^(-2))^3 - (x*A(x))^(-1)/(1 - x^(-1))^2 + 0 + x*A(x) + 2*(x*A(x))^2*(1 - x^2) + 3*(x*A(x))^3*(1 - x^3)^2 + 4*(x*A(x))^4*(1 - x^4)^3 + 5*(x*A(x))^5*(1 - x^5)^4 + ... + n*(x*A(x))^n*(1 - x^n)^(n-1) + ...
SPECIFIC VALUES.
A(1/4) = -1.8892616570712410815999763792198265088...
A(1/5) = -1.6334109911560757412636074394753603214...
A(1/6) = -1.4868349923582400870800926746579742411...
We can illustrate the sum in the definition at x = 1/4.
The sum
0 = Sum_{n=-oo..+oo, n<>0} n * 1/4^n * (1 - 1/4^n)^(n-1) * A(1/4)^n
simplifies somewhat to
0 = Sum_{n=-oo..+oo, n<>0} n * (4^n - 1)^(n-1) * A(1/4)^n / 4^(n^2),
which can be split up into parts P and Q.
Let P denote the sum from -oo to -1, which can be written as
P = Sum_{n>1} (-1)^n * n * 4^n / ((4^n - 1)^(n+1) * A(1/4)^n),
and let Q denote the sum from +1 to +oo:
Q = Sum_{n>1} n * (4^n - 1)^(n-1) * A(1/4)^n / 4^(n^2).
Substituting A(1/4) = -1.8892616570712410815999763792198265088... yields
P = 0.237905890404564510234837963872429856... and
Q = -0.237905890404564510234837963872429856...
so that P + Q = 0.
PROG
(PARI) {a(n) = my(A=[-1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=-#A, #A, if(n==0, 0, n * x^n * (1 - x^n +x*O(x^#A) )^(n-1) * Ser(A)^n )), #A)/2 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 03 2022
STATUS
approved