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%I #8 Mar 08 2024 16:05:38
%S 1,1,-2,4,-5,31,-45,-57,-66,1124,116,-8314,-21328,76424,229013,
%T -537885,-3308391,2165589,36968501,16926113,-402914677,-602853397,
%U 3854818970,11099464628,-31321810278,-161502827466,188659370973,2076430792115,118544397194,-23981088656686
%N Expansion of g.f. A(x) = G( x*(1 + 2*x)*G(x) )^(1/2) = G( x^2*(1 + 3*x)*G(x) )^(1/3), where G(x) is the g.f. of A370437.
%H Paul D. Hanna, <a href="/A370438/b370438.txt">Table of n, a(n) for n = 1..501</a>
%e G.f.: A(x) = x + x^2 - 2*x^3 + 4*x^4 - 5*x^5 + 31*x^6 - 45*x^7 - 57*x^8 - 66*x^9 + 1124*x^10 + 116*x^11 - 8314*x^12 - 21328*x^13 + 76424*x^14 + 229013*x^15 + ...
%e where A(x) = G( x*(1 + 2*x)*G(x) )^(1/2) = G( x^2*(1 + 3*x)*G(x) )^(1/3)
%e and G(x) is the g.f. of A370437, which begins
%e G(x) = x - 3*x^3 + 10*x^4 - 15*x^5 + 84*x^6 - 161*x^7 - 174*x^8 + 612*x^9 + 1596*x^10 + 1926*x^11 - 38592*x^12 - 5895*x^13 + 234684*x^14 + ...
%e RELATED SERIES.
%e A(x)^2 = G( x*(1 + 2*x)*G(x) ) = x^2 + 2*x^3 - 3*x^4 + 4*x^5 + 2*x^6 + 36*x^7 + 8*x^8 - 368*x^9 + 207*x^10 + 1674*x^11 + 3699*x^12 + ...
%e A(x)^3 = G( x^2*(1 + 3*x)*G(x) ) = x^3 + 3*x^4 - 3*x^5 + x^6 + 15*x^7 + 39*x^8 + 88*x^9 - 684*x^10 + 36*x^11 + 3514*x^12 + 6807*x^13 + ...
%e A(x)^6 = x^6 + 6*x^7 + 3*x^8 - 16*x^9 + 45*x^10 + 162*x^11 + 321*x^12 - 1044*x^13 - 4257*x^14 + 12694*x^15 + 37275*x^16 + ...
%e where A(x)^6 = G( x*(1 + 2*x)*G(x) )^3 = G( x^2*(1 + 3*x)*G(x) )^2.
%o (PARI) {a(n) = my(A,G, V=[1]); for(i=1, n+1, V = concat(V, 0); G = x*Ser(V);
%o V[#V] = polcoeff( subst(G, x, x^2*(1 + 3*x)*G )^2 - subst(G, x, x*(1 + 2*x)*G )^3, #V+5); ); A = subst(G, x, x*(1 + 2*x)*G )^(1/2); polcoeff(A,n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A370437, A370538.
%K sign
%O 1,3
%A _Paul D. Hanna_, Mar 08 2024
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