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URL: https://oeis.org/A185388

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A185388
E.g.f. A(x), where A(x)=x*exp(A(x))+x*log(1/(1-A(x))).
1
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
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2013.
FORMULA
a(n) = n^(n-2) + (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 > 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 A379932 A381505
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Feb 21 2011
EXTENSIONS
Offset corrected by Vaclav Kotesovec, Jan 24 2014
STATUS
approved