A186448 E.g.f. A(x) satisfies 2*A(x) = x*(1 + A(x) + exp(A(x))).

1, 2, 10, 88, 1144, 19856, 432464, 11348352, 348715392, 12286859008, 488470565632, 21633197775872, 1056315874429952, 56382210082129920, 3266205054434912256, 204097766901573320704, 13684668496370094407680

Offset: 1

Author

Vladimir Kruchinin, Feb 22 2011

Keywords

• nonn

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.

Crossrefs

Cf. A202357.

Programs

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 *)

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;

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