cp's OEIS Frontend

A194006 E.g.f.: exp( exp(3*x) - 2*exp(2*x) + 2*exp(x) - 1 ).

%I A194006 #29 Oct 22 2024 05:40:08
%S A194006 1,1,4,23,149,1122,9895,99017,1095022,13225701,173261337,2444494252,
%T A194006 36889309869,592107098533,10063638652228,180435656257819,
%U A194006 3401248197894641,67204507885768562,1388196723844580331,29907194746593479677,670590559444043372630
%N A194006 E.g.f.: exp( exp(3*x) - 2*exp(2*x) + 2*exp(x) - 1 ).
%H A194006 Robert Israel, <a href="/A194006/b194006.txt">Table of n, a(n) for n = 0..473</a>
%F A194006 a(n) = Sum_{k=1..n} Sum_{j=k..n} Sum_{i=0..floor((j-k)/3)} j!*(-1)^i*binomial(j-3*i-1,k-1)*binomial(k,k-i)*Stirling2(n,j)/k!, n>1, a(0)=1.
%F A194006 a(n) ~ n^(n + 1/2) * exp(-1 + 2*exp(r) - 2*exp(2*r) + exp(3*r) - n) / (r^n * sqrt(exp(r) * r * (2*(1 + r) - 4 * exp(r) * (1 + 2*r) + exp(2*r) * (3 + 9*r)))), where r = LambertW(n)/3 + (4*n^(1/3)/LambertW(n)^(1/3) - 2) / (9*n^(2/3)/LambertW(n)^(2/3) + 9*n^(2/3)/LambertW(n)^(5/3) - 4*n^(1/3)*(3 + 2*LambertW(n))/LambertW(n)^(4/3) + 6/LambertW(n) + 2). - _Vaclav Kotesovec_, Jul 05 2022
%p A194006 E:=exp( exp(3*x) - 2*exp(2*x) + 2*exp(x) - 1 ):
%p A194006 S:= series(E,x,51):
%p A194006 seq(coeff(S,x,n)*n!,n=0..50); # _Robert Israel_, Oct 05 2020
%o A194006 (Maxima) a(n):=if n=0 then 1 else sum(sum(j!*(sum((-1)^i*binomial(j-3*i-1,k-1)* binomial(k,k-i),i,0,(j-k)/3))*stirling2(n,j),j,k,n)/k!,k,1,n);
%K A194006 nonn
%O A194006 0,3
%A A194006 _Vladimir Kruchinin_, Aug 11 2011