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%I #8 Sep 26 2016 19:00:59
%S 1,0,-3,0,85,0,-6111,0,872649,0,-195062395,0,76208072733,0,
%T -12330526252695,0,125980697776559377,0,857710566759117989133,0,
%U 11428318296234746748941925,0,222333914273403535165432496561,0,6242434914385931957857138485252825,0,244888574110309970555770302512462694549,0,13082369513456349871152908238665975845490989,0,930879791318792717095933863751868808486774883065,0
%N E.g.f. A(x) satisfies: inverse of function A(x)*exp(i*A(x)) equals the conjugate, A(x)*exp(-i*A(x)), where i=sqrt(-1).
%C Apart from signs, essentially the same as A276909.
%H Paul D. Hanna, <a href="/A276910/b276910.txt">Table of n, a(n) for n = 1..301</a>
%F E.g.f. A(x) satisfies: A( A(x)*exp(i*A(x)) ) = i*LambertW(-i*x), where LambertW( x*exp(x) ) = x.
%e E.g.f.: A(x) = x - 3*x^3/3! + 85*x^5/5! - 6111*x^7/7! + 872649*x^9/9! - 195062395*x^11/11! + 76208072733*x^13/13! - 12330526252695*x^15/15! + 125980697776559377*x^17/17! + 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! + 222333914273403535165432496561*x^23/23! + 6242434914385931957857138485252825*x^25/25! +...
%e such that Series_Reversion( A(x)*exp(i*A(x)) ) = A(x)*exp(-i*A(x)).
%e RELATED SERIES.
%e A(x)*exp(i*A(x)) = x + 2*I*x^2/2! - 6*x^3/3! - 28*I*x^4/4! + 180*x^5/5! + 1446*I*x^6/6! - 13888*x^7/7! - 156472*I*x^8/8! + 2034000*x^9/9! + 29724490*I*x^10/10! - 476806176*x^11/11! - 8502508884*I*x^12/12! + 174802753216*x^13/13! + 3768345692398*I*x^14/14! - 63300353418240*x^15/15! - 1386349221087856*I*x^16/16! + 149879079531401472*x^17/17! +...+ A276911(n)*i^(n-1)*x^n/n! +...
%e exp(i*A(x)) = 1 + I*x - x^2/2! - 4*I*x^3/3! + 13*x^4/4! + 116*I*x^5/5! - 661*x^6/6! - 8632*I*x^7/7! + 70617*x^8/8! + 1247248*I*x^9/9! - 13329001*x^10/10! - 285675776*I*x^11/11! + 3782734693*x^12/12! + 107823153088*I*x^13/13! - 1685127882621*x^14/14! - 28683829833856*I*x^15/15! + 574020572798641*x^16/16! + 133507199865641216*I*x^17/17! +...+ A276912(n)*i^(n-1)*x^n/n! +...
%e Also, A( A(x)*exp(i*A(x)) ) = i*LambertW(-i*x), which begins:
%e A( A(x)*exp(i*A(x)) ) = x + 2*I*x^2/2! - 9*x^3/3! - 64*I*x^4/4! + 625*x^5/5! + 7776*I*x^6/6! - 117649*x^7/7! - 2097152*I*x^8/8! +...+ -n^(n-1)*(-i)^(n-1)*x^n/n! +...
%o (PARI) {a(n) = my(V=[1],A=x); for(i=1,n\2+1, V = concat(V,[0,0]); A = sum(m=1,#V,V[m]*x^m/m!) +x*O(x^#V); V[#V] = -(#V)!/2 * polcoeff( subst( A*exp(I*A), x, A*exp(-I*A) ),#V) );V[n]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A276909, A276911, A276912, A179270.
%K sign
%O 1,3
%A _Paul D. Hanna_, Sep 22 2016
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