%I #11 Nov 14 2017 22:40:24
%S 1,-1,-7,5,87,-189,-1941,6515,49795,-229867,-1343239,8320303,36790861,
%T -305098413,-989510467,11255950785,24982028631,-415990506193,
%U -537407696757,15351256446099,6424844980489,-564158722662119,245084502553583,20595643407110741,-26734732721361955,-744981949442089559,1643062426565260297,26621246008843146659,-85112137855646858055,-936300170489142867171,4053227035994699790679,32249453191213868198529
%N G.f. A(x) satisfies: Product_{n=-oo..+oo} 1 + x^n*(1 + A(x)^n)^n = 4.
%H Paul D. Hanna, <a href="/A294474/b294474.txt">Table of n, a(n) for n = 1..200</a>
%F G.f. A(x) satisfies: P(x) * Q(x) = 4 where
%F P(x) = Product_{n>=0} ( 1 + x^n*(1 + A(x)^n)^n ),
%F Q(x) = Product_{n>=1} ( 1 + A(x)^(n^2)/(x + x*A(x)^n)^n ).
%e G.f.: A(x) = x - x^2 - 7*x^3 + 5*x^4 + 87*x^5 - 189*x^6 - 1941*x^7 + 6515*x^8 + 49795*x^9 - 229867*x^10 - 1343239*x^11 + 8320303*x^12 + 36790861*x^13 - 305098413*x^14 - 989510467*x^15 + 11255950785*x^16 + 24982028631*x^17 - 415990506193*x^18 - 537407696757*x^19 + 15351256446099*x^20 + 6424844980489*x^21 - 564158722662119*x^22 + 245084502553583*x^23 + 20595643407110741*x^24 - 26734732721361955*x^25 +...
%e such that P(x) * Q(x) = 4 where
%e P(x) = 2 * (1 + x*(1 + A(x))) * (1 + x^2*(1 + A(x)^2)^2) * (1 + x^3*(1 + A(x)^3)^3) * (1 + x^4*(1 + A(x)^4)^4) * (1 + x^5*(1 + A(x)^5)^5) *...
%e Q(x) = (1 + A(x)/(x + x*A(x))) * (1 + A(x)^4/(x + x*A(x)^2)^2) * (1 + A(x)^9/(x + x*A(x)^3)^3) * (1 + A(x)^16/(x + x*A(x)^4)^4) *...
%e Explicitly,
%e P(x) = 2 + 2*x + 4*x^2 + 2*x^3 - 4*x^4 + 12*x^5 + 120*x^6 - 352*x^7 - 2980*x^8 + 11680*x^9 + 78820*x^10 - 402092*x^11 - 2127062*x^12 + 14447232*x^13 + 57933462*x^14 - 527035730*x^15 - 1535125698*x^16 + 19361173486*x^17 + 37513015102*x^18 - 712839675304*x^19 - 740477725406*x^20 + 26211270910440*x^21 + 4739879806196*x^22 - 959773087051380*x^23 + 640308620069676*x^24 + 34904312707737186*x^25 +...
%e Q(x) = 2 - 2*x - 2*x^2 + 4*x^3 + 6*x^4 - 28*x^5 - 100*x^6 + 642*x^7 + 2322*x^8 - 18850*x^9 - 56244*x^10 + 594794*x^11 + 1327778*x^12 - 19779660*x^13 - 28527454*x^14 + 677346604*x^15 + 440164160*x^16 - 23558817688*x^17 + 3440836740*x^18 + 825188815716*x^19 - 792149118816*x^20 - 28930022822608*x^21 + 52487049980348*x^22 + 1010034689733148*x^23 - 2767421447992010*x^24 - 34946816786828024*x^25 +...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); G = x*Ser(A); A[#A] = -Vec(prod(m=-#A-1, #A+1, 1 + x^m*(1 + G^m)^m ))[#A]/2);A[n]}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A293603.
%K sign
%O 1,3
%A _Paul D. Hanna_, Nov 14 2017