%I #7 Jun 10 2022 11:31:54
%S 1,-1,-4,44,316,-22695,-769536,156937802,30299780744,-18827264809946,
%T -17187430890378027,37887447329364481223,148620374587239353630657,
%U -1249806569497062808351943525,-20168103472406206381500342351035,666759209181977763318463790517458280
%N G.f. A(x) satisfies: 1 = Sum_{n=-oo..+oo} (x + x^n)^n * (-2*A(x))^(n*(n-1)/2).
%H Paul D. Hanna, <a href="/A354646/b354646.txt">Table of n, a(n) for n = 0..100</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F (1) 1 = Sum_{n=-oo..+oo} x^n * (1 + x^(n-1))^n * (-2*A(x))^(n*(n-1)/2).
%F (2) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 + x^(n+1))^n * (-2*A(x))^(n*(n+1)/2).
%e G.f.: A(x) = 1 - x - 4*x^2 + 44*x^3 + 316*x^4 - 22695*x^5 - 769536*x^6 + 156937802*x^7 + 30299780744*x^8 - 18827264809946*x^9 - 17187430890378027*x^10 ++-- ...
%e such that
%e B(x) = Sum_{n>=1} x^n * (1 + x^(n-1))^n * (-2*A(x))^(n*(n-1)/2)
%e and
%e B(x) = -Sum_{n>=1} x^(n^2) / (1 + x^(n+1))^n * (-2*A(x))^(n*(n+1)/2),
%e where
%e B(x) = 2*x - 2*x^2 - 10*x^3 + 98*x^4 + 618*x^5 - 45552*x^6 - 1538490*x^7 + 313926892*x^8 + 60600533658*x^9 - 37654860921240*x^10 - 34374918573912040*x^11 ++-- ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff(sum(m=-#A,#A, (x + x^m)^m * (-2*Ser(A))^(m*(m-1)/2) ),#A)/2);H=A;A[n+1]}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A260116.
%K sign
%O 0,3
%A _Paul D. Hanna_, Jun 07 2022