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A276909 E.g.f. A(x) satisfies: Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)).

%I A276909 #23 Oct 11 2016 18:50:17
%S A276909 1,0,3,0,85,0,6111,0,872649,0,195062395,0,76208072733,0,
%T A276909 12330526252695,0,125980697776559377,0,-857710566759117989133,0,
%U A276909 11428318296234746748941925,0,-222333914273403535165432496561,0,6242434914385931957857138485252825,0,-244888574110309970555770302512462694549,0,13082369513456349871152908238665975845490989,0,-930879791318792717095933863751868808486774883065,0
%N A276909 E.g.f. A(x) satisfies: Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)).
%C A276909 It appears that a(6*k+5) = 1 (mod 3) for k>=0 with a(n) = 0 (mod 3) elsewhere.
%C A276909 Apart from signs, essentially the same as A276910.
%C A276909 E.g.f. A(x) equals the series reversion of the e.g.f. of A276908.
%H A276909 Paul D. Hanna, <a href="/A276909/b276909.txt">Table of n, a(n) for n = 1..301</a>
%F A276909 E.g.f. A(x) satisfies:
%F A276909 (1) A( A(x)*exp(A(x)) ) = -LambertW(-x),
%F A276909 (2) A( A(x)*exp(-A(x)) ) = LambertW(x),
%F A276909 where LambertW( x*exp(x) ) = x.
%F A276909 (3) Series_Reversion( A( x*exp(x) ) ) = A( x*exp(-x) ).
%e A276909 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 A276909 such that Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)).
%e A276909 RELATED SERIES.
%e A276909 A(x)*exp(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1446*x^6/6! + 13888*x^7/7! + 156472*x^8/8! + 2034000*x^9/9! + 29724490*x^10/10! + 476806176*x^11/11! + 8502508884*x^12/12! + 174802753216*x^13/13! + 3768345692398*x^14/14! + 63300353418240*x^15/15! + 1386349221087856*x^16/16! + 149879079531401472*x^17/17! +...+ A276911(n)*x^n/n! +...
%e A276909 exp(A(x)) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 116*x^5/5! + 661*x^6/6! + 8632*x^7/7! + 70617*x^8/8! + 1247248*x^9/9! + 13329001*x^10/10! + 285675776*x^11/11! + 3782734693*x^12/12! + 107823153088*x^13/13! + 1685127882621*x^14/14! + 28683829833856*x^15/15! + 574020572798641*x^16/16! + 133507199865641216*x^17/17! +...+ A276912(n)*x^n/n! +...
%e A276909 Also,  A( A(x)*exp(A(x)) ) = -LambertW(-x), which begins:
%e A276909 A( A(x)*exp(A(x)) ) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...
%o A276909 (PARI) {a(n) = my(A=x +x*O(x^n));
%o A276909 for(i=1,n, A = A + (x - subst(A*exp(A),x,A*exp(-A)))/2); n!*polcoeff(A,n)}
%o A276909 for(n=1, 30, print1(a(n), ", "))
%o A276909 (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(A), x, A*exp(-A) ), #V) ); V[n]}
%o A276909 for(n=1, 30, print1(a(n), ", "))
%Y A276909 Cf. A276910, A276911, A276912, A276913, A179270, A276908.
%K A276909 sign
%O A276909 1,3
%A A276909 _Paul D. Hanna_, Sep 26 2016