A365032 -id:A365032 - cp's OEIS frontend

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

Offset: 0

Paul D. Hanna, Oct 07 2003

nonn

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.

Seiichi Manyama, Table of n, a(n) for n = 0..364
Index entries for reversions of series

Cf. A143768, A362771.

Cf. A365031, A365032.

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

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

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
E.g.f.: (1/x) * Series_Reversion( x*exp(-x*(1 + x)) ). - Seiichi Manyama, Sep 23 2024

A365031 E.g.f. satisfies A(x) = exp(x * A(x) * (1 + x * A(x))^2).

Original entry on oeis.org

1, 1, 7, 70, 1085, 22176, 569107, 17583616, 636085305, 26383168000, 1234691104031, 64368785424384, 3699873561469813, 232476344504965120, 15853643565560296875, 1166213594266747273216, 92052000392983157418353, 7760655405804462332903424

Offset: 0

Seiichi Manyama, Aug 17 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..352
Index entries for reversions of series

Cf. A088695, A365032.

Cf. A364939.

Array[#!*Sum[ (# + 1)^(k - 1)*Binomial[2 k, # - k]/k!, {k, 0, #}] &, 18, 0] (* Michael De Vlieger, Aug 18 2023 *)

a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*k, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*k,n-k)/k!.

E.g.f.: (1/x) * Series_Reversion( x*exp(-x*(1 + x)^2) ). - Seiichi Manyama, Sep 23 2024

cp's OEIS Frontend

A088695 E.g.f. satisfies A(x) = f(x*A(x)), where f(x) = exp(x+x^2).

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