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A088695 revision #24

A088695

E.g.f. satisfies A(x) = f(x*A(x)), where f(x) = exp(x+x^2).

1, 1, 5, 40, 485, 7776, 156457, 3788800, 107414505, 3491200000, 128019454541, 5229222395904, 235490648957005, 11592449531084800, 619331166211640625, 35691050995648823296, 2206955604752999720273, 145757527499874820423680, 10240455593560436925898645

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OFFSET

0,3

COMMENTS

Radius of convergence of A(x): r = (1/2)*exp(-3/4) = 0.23618..., where A(r) = exp(3/4) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..364

Index entries for reversions of series

FORMULA

a(n) = n! * [x^n] exp(x+x^2)^(n+1)/(n+1).

a(n) = n! * Sum_{k=floor(n/2)..n} binomial(k,n-k)*(n+1)^(k-1)/k!. - Vladimir Kruchinin, Aug 04 2011

a(n) ~ 2^(n+1/2) * n^(n-1) / (sqrt(3) * exp(n/4 - 3/4)). - Vaclav Kotesovec, Jan 24 2014

MATHEMATICA

Table[n!*SeriesCoefficient[(E^(x+x^2))^(n+1)/(n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2014 *)

PROG

(PARI) a(n)=n!*polcoeff(exp(x+x^2)^(n+1)+x*O(x^n), n, x)/(n+1)

CROSSREFS