OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..365
FORMULA
a(n) = Sum_{k=0..n} C(n,k) * (2*(n-k) + 1)^(k-1) * k^(n-k).
E.g.f.: A(x) = F(x)^(1/2) where F(x) = e.g.f. of A161566.
E.g.f.: A(x) = exp(x*G(x)) where G(x) = e.g.f. of A161565.
a(n) ~ n^(n-1) / (2*exp(n)*r^(n+1/2)), where r = 0.256263163133653382... is the root of the equation 1/LambertW(1/r) = -log(2*r^2) - LambertW(1/r). - Vaclav Kotesovec, Feb 28 2014
EXAMPLE
MATHEMATICA
Flatten[{1, Table[Sum[Binomial[n, k] * (2*(n-k) + 1)^(k-1) * k^(n-k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 28 2014 *)
PROG
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(2*(n-k)+1)^(k-1)*k^(n-k))}
(PARI) {a(n)=local(A=1+x); for(i=0, n, A=exp(x*exp(x*A^2+O(x^n)))); n!*polcoeff(A, n, x)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 14 2009
STATUS
approved