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%I #9 Jun 18 2016 15:07:57
%S 1,-2,6,-40,320,-2976,35392,-538112,9931392,-211790080,5059784576,
%T -132643057152,3761875287040,-114501941915648,3725395402721280,
%U -129324055589257216,4786638435256696832,-188785468724361560064,7922155381738193944576,-352740315643746941665280,16603695476218208847691776,-822951583413551750366298112,42792449844854211313594597376,-2327246576567999111735900897280,132052357036729088907927420928000
%N E.g.f. A(x) satisfies: A( A(x)^2 ) = x^2 * exp(-2*x).
%F E.g.f. equals the series reversion of the e.g.f. of A274275.
%F Given e.g.f. A(x), let B(x) satisfy A(B(x)) = x, then
%F (1) A(x) = sqrt( B( x^2*exp(-2*x) ) ).
%F (2) A(x^2) = B(x)^2 * exp(-2*B(x)).
%e E.g.f.: A(x) = x - 2*x^2/2! + 6*x^3/3! - 40*x^4/4! + 320*x^5/5! - 2976*x^6/6! + 35392*x^7/7! - 538112*x^8/8! + 9931392*x^9/9! - 211790080*x^10/10! + 5059784576*x^11/11! - 132643057152*x^12/12! + 3761875287040*x^13/13! - 114501941915648*x^14/14! + 3725395402721280*x^15/15! - 129324055589257216*x^16/16! +...
%e where A( A(x)^2 ) = x^2 * exp(-2*x).
%e RELATED SERIES.
%e Let B(x) be the series reversion of the e.g.f. A(x), which begins
%e B(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 400*x^5/5! + 4656*x^6/6! + 62944*x^7/7! + 1046144*x^8/8! + 20274048*x^9/9! + 438238720*x^10/10! + 10529132416*x^11/11! + 280439144448*x^12/12! + 8185848206848*x^13/13! + 259202608222208*x^14/14! +...+ A274275(n)*x^n/n! +...
%e then A(x) = sqrt( B( x^2*exp(-2*x) ) )
%e and A(x^2) = B(x)^2 * exp(-2*B(x)).
%e A(x)^2 = 2*x^2/2! - 12*x^3/3! + 72*x^4/4! - 640*x^5/5! + 6960*x^6/6! - 85344*x^7/7! + 1226624*x^8/8! - 21007872*x^9/9! + 422254080*x^10/10! - 9724042240*x^11/11! + 250998494208*x^12/12! +...
%e where A(x)^2 = B( x^2*exp(-2*x) ) such that B(A(x)) = x.
%o (PARI) /* From A(x) = sqrt( B( x^2*exp(-2*x) ) ) where A(B(x)) = x */
%o {a(n) = my(A=x,B=x); for(i=1, n, B = serreverse(A +x*O(x^n)); A = sqrt( subst(B, x, x^2*exp(-2*x +x*O(x^n))) ) ); n!*polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%o (PARI) /* As the series reversion of the e.g.f. of A274275 */
%o {a(n) = my(B=x); for(i=1, n, B = serreverse( sqrt( subst(B, x, x^2*exp(-2*x +x*O(x^n))) ) ) ); n!*polcoeff(serreverse(B), n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A274275.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jun 18 2016
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