A185386 E.g.f. A(x) = (x*e^A(x) + x*e^(A(x)^2))/2.

0, 1, 1, 6, 32, 330, 3588, 53144, 872176, 17110476, 372514640, 9167850648, 247662013440, 7353997830296, 236810884471168, 8244887609371200, 308077011732259328, 12309365766783226512, 523406306762717739264, 23606320303325424943328, 1125382002581625789399040

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..390

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011.

Formula

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

a(n) ~ sqrt((s*(1-2*s))/(-1+2*s-6*s^2+4*s^3)) * n^(n-1) / (exp(n) * r^n), where s = 0.7899321073754988125... is the root of the equation s*(exp(s) + 2*exp(s^2)*s) = exp(s) + exp(s^2), and r = 2*s/(exp(s) + exp(s^2)) = 0.3882100419285789435... - Vaclav Kotesovec, Jan 23 2014

Mathematica

CoefficientList[InverseSeries[Series[(2*x)/(E^x + E^x^2), {x, 0, 20}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2014 *)
Flatten[{0, Table[n!*(Sum[Binomial[n, m]*Sum[m^(i-m)*(n-m)^((n+m-i-1)/2)*(1+(-1)^(n+m-i-1))/((i-m)!*((n+m-i-1)/2)!), {i, m, n+m-1}], {m, 1, n-1}]/2+n^(n-1)/(n-1)!+n^((n-1)/2)*(1+(-1)^(n-1))/(2*((n-1)/2)!))/(n*2^n), {n, 1, 20}]}] (* Vaclav Kotesovec after Vladimir Kruchinin, Jan 23 2014 *)

Prog

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

Crossrefs

Sequence in context: A326985 A350485 A350109 * A368287 A276351 A135538

Adjacent sequences: A185383 A185384 A185385 * A185387 A185388 A185389

Keyword

nonn

Author

Vladimir Kruchinin, Feb 20 2011

Extensions

Prepended missing a(0)=0 from Vaclav Kotesovec, Jan 23 2014

Status

approved