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A193341
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E.g.f. satisfies: A(A(x)) = x*exp(A(x)), where A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)).
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2
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1, 2, 6, 16, 0, -144, 5488, 47104, -2799360, -29427200, 3293554176, 40830142464, -7642645477376, -109489995819008, 31826754503424000, 518027268557111296, -221570477108873330688, -4041287223180417957888, 2438941389381370203996160, 49292069262802363796684800
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OFFSET
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1,2
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COMMENTS
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It is surprising that A(x)*exp(-x/2)/x is an even function (cf. A193342).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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) +...
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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