A185388 E.g.f. A(x), where A(x)=x*exp(A(x))+x*log(1/(1-A(x))).

1, 2, 10, 87, 1111, 18915, 404401, 10430994, 315487369, 10954371241, 429594068481, 18783615744828, 906140556605065, 47812585715846653, 2739407974494036119, 169373948577368957236, 11240688061343547469681, 797035895999061505391793, 60135437182098854042617645, 4810429394481317964702524924

Offset

1,2

Comments

A(x) is E.g.f. for a(n)/n. - Vaclav Kotesovec, Jan 24 2014

Links

G. C. Greubel, Table of n, a(n) for n = 1..360

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2013.

Formula

a(n) = (1/n)*Sum_{m=1..(n-1)} m!*binomial(n,m)*Sum_{i=2*m..(n+m-1)} binomial(n-1,i-m)*Stirling1(i-m,m)*(-1)^(i-2*m)*(n-m)^(n+m-i-1))) + n^(n-2), n > 0.

a(n) ~ (1-s)*sqrt(s/(1-(2-r)*s+s^2)) * n^(n-2) / (exp(n) * r^n), where s = 0.6276084459806374335... is the root of the equation (exp(s) + 1/(1-s))*s = exp(s) + log(1/(1-s)), and r = 1/(exp(s) + 1/(1-s)) = 0.2193717986293565123... - Vaclav Kotesovec, Jan 24 2014

Mathematica

Table[(Rest[CoefficientList[InverseSeries[Series[x/(E^x + Log[(1 - x)^(-1)]), {x, 0, 20}], x], x] * Range[0, 20]!])[[n]]/n, {n, 1, 20}] (* Vaclav Kotesovec, Jan 24 2014 *)
Table[1/n*Sum[m!*Binomial[n, m]*Sum[Binomial[n-1, i-m]*StirlingS1[i-m, m]*(-1)^(i-2*m)*(n-m)^(n+m-i-1), {i, 2*m, n+m-1}], {m, 1, n-1}]+n^(n-2), {n, 1, 20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Jan 24 2014 *)

Prog

(Maxima)
a(n):=1/n*sum(m!*binomial(n, m)*sum(binomial(n-1, i-m)*stirling1(i-m, m)*(-1)^(i-2*m)*(n-m)^(n+m-i-1), i, 2*m, n+m-1), m, 1, n-1)+n^(n-2);

Crossrefs

Sequence in context: A355083 A295836 A245496 * A245009 A354240 A306404

Adjacent sequences: A185385 A185386 A185387 * A185389 A185390 A185391

Keyword

nonn

Author

Vladimir Kruchinin, Feb 21 2011

Extensions

Offset corrected by Vaclav Kotesovec, Jan 24 2014

Status

approved