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 A139702 G.f. satisfies: x = A( x + A(x)^2 ). 13
 1, -1, 4, -24, 178, -1512, 14152, -142705, 1528212, -17211564, 202460400, -2474708496, 31310415376, -408815254832, 5495451727376, -75907303147652, 1075685334980240, -15618612118252960, 232102241507321384, -3526880759915999016 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Signed version of A213591. LINKS Paul D. Hanna, Table of n, a(n), n=1..100. FORMULA Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then G(x) = G(G(x)) - x^2 = g.f. of A138740. G.f. satisfies: A(x) = x*G(-A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411. G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with: A = 1 - x*B^2; B = A - x*C^2; C = B - x*D^2; D = C - x*E^2; E = D - x*F^2; ... G.f. satisfies: A(x) = x*exp( Sum_{n>=0} (-1)^(n+1)*[d^n/dx^n A(x)^(2n+2)/x]/(n+1)! ). [Paul D. Hanna, Dec 18 2010] EXAMPLE G.f.: A(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-... A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-... where A(x + A(x)^2) = x. Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then: G(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... and G(G(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... so that G(x) = G(G(x)) - x^2 = g.f. of A138740. Logarithmic series: log(A(x)/x) = -A(x)^2/x + [d/dx A(x)^4/x]/2! - [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! -+... PROG (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(A, n))} (PARI) /* n-th Derivative: */ {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} /* G.f.: [Paul D. Hanna, Dec 18 2010] */ {a(n)=local(A=x-x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=0, n, (-1)^(m+1)*Dx(m, A^(2*m+2)/x)/(m+1)!)+x*O(x^n))); polcoeff(A, n)} CROSSREFS Cf. A138740, A212411, A213591. Cf. A088714, A088717, A091713, A120971, A140094, A140095. Cf. A143426, A087949, A143435, A182969. Sequence in context: A000309 A112914 A007846 * A213591 A243689 A168452 Adjacent sequences:  A139699 A139700 A139701 * A139703 A139704 A139705 KEYWORD sign AUTHOR Paul D. Hanna, Apr 30 2008, May 20 2008 STATUS approved

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