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Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).
3

%I #11 May 06 2024 05:37:05

%S -1,0,6,81,1620,45765,1671678,74794671,3958829640,241898775273,

%T 16756621904970,1297547591499819,111065107263415308,

%U 10412999996499836541,1061234184094567585326,116812280111404106348415,13810631408232372091755792,1745470697932523785587735249

%N Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = Sum_{k=0..n} (3*k-1)^(k-1) * binomial(n,k).

%F G.f.: Sum_{k>=0} (3*k-1)^(k-1) * x^k / (1-x)^(k+1).

%F a(n) ~ 3^(n-1) * n^(n-1) * exp((exp(-1) - 1)/3). - _Vaclav Kotesovec_, May 06 2024

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x+lambertw(-3*x)/3)))

%o (PARI) a(n) = sum(k=0, n, (3*k-1)^(k-1)*binomial(n, k));

%Y Cf. A088957, A360193, A372315, A372316, A372320.

%K sign

%O 0,3

%A _Seiichi Manyama_, Apr 27 2024