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%I #18 Oct 05 2016 02:46:25
%S 1,0,3,0,-35,0,6111,0,-3015207,0,3457389595,0,-7910176435083,0,
%T 32652618744201015,0,-225992449753641748943,0,
%U 2477459751096859267509171,0,-41090881423264757483386565235,0,992851798453466404257942193460239,0,-33857339246997857308988305386104611575,0,1586206583926227307173185697414192414735051,0,-99763501980273385738989314186327124186627104987,0
%N E.g.f. A(x) satisfies: Series_Reversion( A(x)*exp(x) ) = A(x)*exp(-x).
%C It appears that a(6*k+5) = 1 (mod 3) for k>=0 with a(n) = 0 (mod 3) elsewhere.
%H Paul D. Hanna, <a href="/A276913/b276913.txt">Table of n, a(n) for n = 1..151</a>
%F E.g.f. A(x) also satisfies:
%F (1) A( A(x)*exp(x) ) = x*exp( A(x)*exp(x) ).
%F (2) A( A(x)*exp(-x) ) = x*exp( -A(x)*exp(-x) ).
%e E.g.f.: A(x) = x + 3*x^3/3! - 35*x^5/5! + 6111*x^7/7! - 3015207*x^9/9! + 3457389595*x^11/11! - 7910176435083*x^13/13! + 32652618744201015*x^15/15! - 225992449753641748943*x^17/17! + 2477459751096859267509171*x^19/19! - 41090881423264757483386565235*x^21/21! + 992851798453466404257942193460239*x^23/23! - 33857339246997857308988305386104611575*x^25/25! +...
%e RELATED SERIES.
%e By definition, Series_Reversion( A(x)*exp(x) ) = A(x)*exp(-x), where
%e A(x)*exp(x) = x + 2*x^2/2! + 6*x^3/3! + 16*x^4/4! - 144*x^6/6! + 5488*x^7/7! + 47104*x^8/8! - 2799360*x^9/9! - 29427200*x^10/10! + 3293554176*x^11/11! + 40830142464*x^12/12! - 7642645477376*x^13/13! - 109489995819008*x^14/14! + 31826754503424000*x^15/15! +...+ A193341(n)*x^n/n! +...
%e A(x)*exp(-x) = x - 2*x^2/2! + 6*x^3/3! - 16*x^4/4! + 144*x^6/6! + 5488*x^7/7! - 47104*x^8/8! - 2799360*x^9/9! +...+ (-1)^(n-1)*A193341(n)*x^n/n! +...
%e Also, A( A(x)*exp(x) ) = x*exp( A(x)*exp(x) ), where
%e A( A(x)*exp(x) ) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 325*x^5/5! + 2046*x^6/6! + 14749*x^7/7! + 166664*x^8/8! + 1855305*x^9/9! - 8673830*x^10/10! - 380002799*x^11/11! + 33613835388*x^12/12! + 913029698893*x^13/13! - 91462474379626*x^14/14! - 2893000394547675*x^15/15! + 452208618208709776*x^16/16! +...
%e exp( A(x)*exp(x) ) = 1 + x + 3*x^2/2! + 13*x^3/3! + 65*x^4/4! + 341*x^5/5! + 2107*x^6/6! + 20833*x^7/7! + 206145*x^8/8! - 867383*x^9/9! - 34545709*x^10/10! + 2801152949*x^11/11! + 70233053761*x^12/12! - 6533033884259*x^13/13! - 192866692969845*x^14/14! + 28263038638044361*x^15/15! +...
%e Also,
%e A'( A(x)*exp(-x) ) * exp( A(x)*exp(-x) ) = exp(x)/(A'(x) - A(x)) - x, or
%e x*A'( A(x)*exp(-x) ) / A( A(x)*exp(-x) ) = exp(x)/(A'(x) - A(x)) - x.
%e The series reversion begins:
%e Series_Reversion( A(x) ) = x - 3*x^3/3! + 125*x^5/5! - 19551*x^7/7! + 8072217*x^9/9! - 7563307675*x^11/11! + 14604702539349*x^13/13! - 53272560312696375*x^15/15! + 338351296939319691953*x^17/17! +...
%o (PARI) {a(n) = my(A = x +x*O(x^n));for(i=1,n, A = A + (x - subst(A*exp(x +x*O(x^n)),x, A*exp(-x +x*O(x^n))))/2); n!*polcoeff(A,n)}
%o for(n=1, 31, print1(a(n), ", "))
%Y Cf. A276909, A276908, A276912, A193341.
%K sign
%O 1,3
%A _Paul D. Hanna_, Oct 01 2016
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