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%I #6 Jun 16 2016 18:29:39
%S 1,1,1,4,77,736,2077,22912,1197625,23597056,350173241,7161708544,
%T 236337969925,6751323455488,122041278706453,3799979465506816,
%U 298712815532930033,10872130692620222464,-18153139467375673487,-513247768690867306496,1172597577739561586096701,53608628175847428085252096,-748272864671493582192607219,-39715579516869644288006291456,7586072261976188853665242247977
%N E.g.f. A(x) satisfies: A( sqrt( A(x^2*exp(-x)) ) ) = x, where A(x) = Sum_{n>=1} a(n)*x^n/(2^(n-1)*(n-1)!).
%F E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / (2^(n-1)*(n-1)!) satisfies:
%F (1) A( sqrt( A(x^2*exp(x)) ) ) = -2*LambertW(-x/2*exp(x/2)).
%F (2) A(x) = Series_Reversion( sqrt( A(x^2*exp(-x)) ) ).
%e E.g.f. A(x) = x + x^2/2 + x^3/(2^2*2!) + 4*x^4/(2^3*3!) + 77*x^5/(2^4*4!) + 736*x^6/(2^5*5!) + 2077*x^7/(2^6*6!) + 22912*x^8/(2^7*7!) + 1197625*x^9/(2^8*8!) + 23597056*x^10/(2^9*9!) + 350173241*x^11/(2^10*10!) + 7161708544*x^12/(2^11*11!) + 236337969925*x^13/(2^12*12!) + 6751323455488*x^14/(2^13*13!) + 122041278706453*x^15/(2^14*14!) +...
%e such that: A( sqrt( A(x^2*exp(-x)) ) ) = x.
%e Written with reduced fraction coefficients,
%e A(x) = x + 1/2*x^2 + 1/8*x^3 + 1/12*x^4 + 77/384*x^5 + 23/120*x^6 + 2077/46080*x^7 + 179/5040*x^8 + 239525/2064384*x^9 + 823/6480*x^10 + 350173241/3715891200*x^11 + 109279/1247400*x^12 + 9453518797/78479622144*x^13 + 206034041/1556755200*x^14 + 122041278706453/1428329123020800*x^15 +...
%e Also, A( sqrt( A(x^2*exp(x)) ) ) = -2*LambertW(-x/2*exp(x/2)) where
%e A( sqrt( A(x^2*exp(x)) ) ) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1476*x^6/6! +...+ A100526(n)*x^n/n! +...
%o (PARI) {a(n) = my(A=x); for(i=1,n, A = serreverse( sqrt( subst(A,x,x^2*exp(-x +x*O(x^n))) ) ) ); (n-1)!*2^(n-1) * polcoeff(A,n)}
%o for(n=1,30, print1(a(n),", "))
%Y Cf. A100526.
%K sign
%O 1,4
%A _Paul D. Hanna_, Jun 16 2016
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