%I #8 Aug 28 2017 07:58:54
%S 1,1,-1,2,-5,13,-40,126,-409,1363,-4617,15896,-55444,195480,-695636,
%T 2495118,-9011281,32741839,-119601339,438968354,-1618006837,
%U 5986803522,-22229028994,82798248894,-309299225632,1158483827048,-4349740078410,16368842606820,-61727972554068,233233590724532,-882851162632794,3347489178793192,-12712739206990305
%N G.f. A(x) satisfies: A( x*A(x)^2 - x^2*A(x) ) = x^4.
%H Paul D. Hanna, <a href="/A291614/b291614.txt">Table of n, a(n) for n = 1..300</a>
%F a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 3.98431348330228671611917... and c = 0.0351241010191430532... - _Vaclav Kotesovec_, Aug 28 2017
%e G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 5*x^5 + 13*x^6 - 40*x^7 + 126*x^8 - 409*x^9 + 1363*x^10 - 4617*x^11 + 15896*x^12 - 55444*x^13 + 195480*x^14 - 695636*x^15 + 2495118*x^16 - 9011281*x^17 + 32741839*x^18 - 119601339*x^19 + 438968354*x^20 +...
%e such that A( x*A(x)^2 - x^2*A(x) ) = x^4.
%e RELATED SERIES.
%e x*A(x)^2 - x^2*A(x) = x^4 - x^8 + 3*x^12 - 12*x^16 + 55*x^20 - 272*x^24 +...
%e Let Ai(x) denote the series reversion of A(x), then
%e Ai(x) = x - x^2 + 3*x^3 - 12*x^4 + 55*x^5 - 272*x^6 + 1418*x^7 - 7674*x^8 + 42703*x^9 - 242802*x^10 + 1404430*x^11 - 8238084*x^12 + 48888198*x^13 - 292986110*x^14 + 1770676784*x^15 - 10779271798*x^16 + 66038787055*x^17 - 406856161566*x^18 + 2519083078066*x^19 - 15666603475448*x^20 +...
%e where Ai( Ai(x)^4 ) = x^2*Ai(x) - x*Ai(x)^2.
%o (PARI) {a(n) = my(A=[1, 1,-1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = polcoeff(x^4 - subst(F, x, x*F^2 - x^2*F), #A+2) ); A[n]}
%o for(n=1, 40, print1(a(n), ", "))
%K sign
%O 1,4
%A _Paul D. Hanna_, Aug 27 2017