OFFSET
1,2
COMMENTS
A(x) is e.g.f. for a(n)/(2^(n-1)/n). - Vaclav Kotesovec, Jan 26 2014
LINKS
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
FORMULA
a(n+1) = (n!/(2*n+2)) * (Sum_{m=1..n} binomial(n+1, m) * (Sum_{i=0..n} (m^i/i!) * binomial(n-m+1, n-i))) + (n+1)^(n-1)/2 + n!/2.
a(n) ~ n^(n-2) * (1+c)^(n+1) / (2 * c^n * exp(n)), where c = LambertW(exp(-1)) = 0.278464542761... (see A202357). - Vaclav Kotesovec, Jan 26 2014
MATHEMATICA
Table[(Rest[CoefficientList[InverseSeries[Series[(2*x)/(1+E^x+x), {x, 0, 20}], x], x]*Range[0, 20]!])[[n]] * (2^(n-1)/n), {n, 1, 20}] (* Vaclav Kotesovec, Jan 26 2014 *)
Table[n!/(2*n+2)*Sum[Binomial[n+1, m]*Sum[m^(i)/i!*Binomial[n-m+1, n-i], {i, 0, n}], {m, 1, n}]+(n+1)^(n-1)/2+n!/2, {n, 0, 20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Jan 26 2014 *)
PROG
(Maxima)
a(n):=n!/(2*n+2)*sum(binomial(n+1, m)*sum(m^(i)/i!*binomial(n-m+1, n-i), i, 0, n), m, 1, n)+(n+1)^(n-1)/2+(n)!/2;
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Feb 22 2011
STATUS
approved