cp's OEIS Frontend

A355672 Expansion of e.g.f. exp(1/(1-x) - exp(x)).

%I A355672 #19 Jul 21 2022 06:45:06
%S A355672 1,0,1,5,26,169,1329,12088,124221,1422307,17947550,247318851,
%T A355672 3693469273,59396067080,1022975862713,18781241965081,366070181352802,
%U A355672 7547972562003093,164113696105503057,3752143293971556144,89976991297720804061,2257905394760969948079
%N A355672 Expansion of e.g.f. exp(1/(1-x) - exp(x)).
%H A355672 Vaclav Kotesovec, <a href="/A355672/b355672.txt">Table of n, a(n) for n = 0..440</a>
%F A355672 a(0) = 1; a(n) = Sum_{k=1..n} (k! - 1) * binomial(n-1,k-1) * a(n-k).
%F A355672 a(n) ~ exp(1/2 - exp(1) + 2*sqrt(n) - n) * n^(n - 1/4) / sqrt(2). - _Vaclav Kotesovec_, Jul 21 2022
%t A355672 nmax = 20; CoefficientList[Series[E^(1/(1-x) - E^x), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 21 2022 *)
%o A355672 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1/(1-x)-exp(x))))
%o A355672 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j!-1)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A355672 Cf. A000262, A000587, A033312, A186755, A352294.
%K A355672 nonn
%O A355672 0,4
%A A355672 _Seiichi Manyama_, Jul 14 2022