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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

Table of n, a(n) for n=1..20.

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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Last modified November 18 12:21 EST 2017. Contains 294891 sequences.