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 A193341 E.g.f. satisfies: A(A(x)) = x*exp(A(x)), where A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)). 2
 1, 2, 6, 16, 0, -144, 5488, 47104, -2799360, -29427200, 3293554176, 40830142464, -7642645477376, -109489995819008, 31826754503424000, 518027268557111296, -221570477108873330688, -4041287223180417957888, 2438941389381370203996160, 49292069262802363796684800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS It is surprising that A(x)*exp(-x/2)/x is an even function (cf. A193342). LINKS FORMULA E.g.f. A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)) also satisfies: (1) A(x) = -A(-x)*exp(x). (2) A( A(x)/exp(x) ) = x. (3) A(-A(-x)) = x. (4) A(x) = x*exp(x/2)*G(x) where G(x) is the even function described by A193342. EXAMPLE E.g.f.: A(x) = x + 2*x^2/(2!*2) + 6*x^3/(3!*4) + 16*x^4/(4!*8) - 144*x^6/(6!*32) + 5488*x^7/(7!*64) + 47104*x^8/(8!*128) - 2799360*x^9/(9!*256) - 29427200*x^10/(10!*512) +... where A(A(x)) = x*exp(A(x)) begins: A(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 22*x^4/4! + 90*x^5/5! + 396*x^6/6! + 1918*x^7/7! + 10830*x^8/8! + 66510*x^9/9! + 325450*x^10/10! +... The series reversion begins: A(x)*exp(-x) = -A(-x) = x - 2*x^2/(2!*2) + 6*x^3/(3!*4) - 16*x^4/(4!*8) + 144*x^6/(6!*32) +... so that the g.f. satisfies: -A(x)/A(-x) = exp(x). The e.g.f. G(x) = A(x)*exp(-x/2)/x is an even function: G(x) = 1 + x^2/(2!*2^2) - 7*x^4/(4!*2^4) + 873*x^6/(6!*2^6) - 335023*x^8/(8!*2^8) + 314308145*x^10/(10!*2^10) +... PROG (PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=A+(x*exp(A+x*O(x^n))-subst(A, x, A))/2); n!*2^(n-1)*polcoeff(A, n)} CROSSREFS Cf. A276913, A193342. Sequence in context: A261726 A181993 A123475 * A009711 A009586 A009487 Adjacent sequences:  A193338 A193339 A193340 * A193342 A193343 A193344 KEYWORD sign AUTHOR Paul D. Hanna, Jul 23 2011 STATUS approved

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