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%I #29 Oct 18 2022 11:38:51
%S -1,-2,-4,-8,-8,-6,40,132,400,504,76,-4960,-18528,-56998,-94176,
%T -58896,617216,2911128,9741760,19739472,21657312,-75073186,-483271024,
%U -1800924184,-4274295720,-6374947674,7150661892,81254492928,345397065128,937137978804,1717431001440
%N a(n) = coefficient of x^n in the power series A(x) such that: 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n, starting with a(0) = -1.
%C Related identity: 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1), which holds when 0 < |x| < 1.
%C 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.
%H Paul D. Hanna, <a href="/A357159/b357159.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n, with a(0) = -1, satisfies the following relations.
%F (1) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n.
%F (2) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n/A(x))^(n-1).
%F (3) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (A(x) - x^n)^(n-1).
%F (4) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) / ( (1 - x^n)^(n+1) * A(x)^n ).
%F (5) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) * A(x)^n / (A(x) - x^n)^(n+1).
%F (6) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) * A(x)^n / (1 - x^n*A(x))^(n+1).
%e 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 + ...
%e such that
%e 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) + ...
%e SPECIFIC VALUES.
%e A(1/4) = -1.8892616570712410815999763792198265088...
%e A(1/5) = -1.6334109911560757412636074394753603214...
%e A(1/6) = -1.4868349923582400870800926746579742411...
%e We can illustrate the sum in the definition at x = 1/4.
%e The sum
%e 0 = Sum_{n=-oo..+oo, n<>0} n * 1/4^n * (1 - 1/4^n)^(n-1) * A(1/4)^n
%e simplifies somewhat to
%e 0 = Sum_{n=-oo..+oo, n<>0} n * (4^n - 1)^(n-1) * A(1/4)^n / 4^(n^2),
%e which can be split up into parts P and Q.
%e Let P denote the sum from -oo to -1, which can be written as
%e P = Sum_{n>1} (-1)^n * n * 4^n / ((4^n - 1)^(n+1) * A(1/4)^n),
%e and let Q denote the sum from +1 to +oo:
%e Q = Sum_{n>1} n * (4^n - 1)^(n-1) * A(1/4)^n / 4^(n^2).
%e Substituting A(1/4) = -1.8892616570712410815999763792198265088... yields
%e P = 0.237905890404564510234837963872429856... and
%e Q = -0.237905890404564510234837963872429856...
%e so that P + Q = 0.
%o (PARI) {a(n) = my(A=[-1]); for(i=1,n, A=concat(A,0);
%o 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]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A291937, A357158.
%K sign
%O 0,2
%A _Paul D. Hanna_, Oct 03 2022
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