%I #18 Sep 25 2024 15:06:31

%S 1,4,43,847,23881,870721,38894653,2055873037,125480383153,

%T 8684069883409,671922832985941,57475677232902589,5385592533714824521,

%U 548596467532888667257,60358911366712739334541,7133453715771227363127301,901261693601873814393568993

%N a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 3^k * k^k.

%H Robert Israel, <a href="/A277456/b277456.txt">Table of n, a(n) for n = 0..333</a>

%F E.g.f.: exp(x)/(1+LambertW(-3*x)).

%F a(n) ~ exp(exp(-1)/3) * 3^n * n^n.

%p f:= n -> 1 + add(binomial(n,k)*3^k*k^k,k=1..n):

%p map(f, [$0..20]); # _Robert Israel_, Oct 30 2016

%t Table[1 + Sum[Binomial[n, k]*3^k*k^k, {k, 1, n}], {n, 0, 20}]

%t CoefficientList[Series[E^x/(1+LambertW[-3*x]), {x, 0, 20}], x] * Range[0, 20]!

%o (PARI) a(n) = 1 + sum(k=1, n, binomial(n,k) * 3^k * k^k); \\ _Michel Marcus_, Oct 30 2016

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(x)/(1+lambertw(-3*x)))) \\ _G. C. Greubel_, Sep 09 2018

%o (Magma) [1] cat [1 + (&+[Binomial(n,k)*3^k*k^k: k in [1..n]]): n in [1..20]]; // _G. C. Greubel_, Sep 09 2018

%Y Cf. A086331, A277454.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Oct 16 2016