%I #10 Dec 02 2023 07:49:02

%S 1,-4,15,-53,178,-575,1809,-5598,17141,-52113,157724,-475997,1433429,

%T -4311364,12958627,-38909601,116831426,-350844883,1051414421,

%U -3160120038,9491592177,-28218244109,86403627444,-255153772169,722619907385,-2772952748516,4627276967623,-17420488524253

%N Expansion of e.g.f. exp(1 - 3*x - exp(x)).

%F G.f. A(x) satisfies: A(x) = 1 - 3*x*A(x) - x * A(x/(1 - x)) / (1 - x).

%F a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k-3)^n / k!.

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

%F a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A000587(k).

%t nmax = 27; CoefficientList[Series[Exp[1 - 3 x - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = -3 a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]

%t Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 27}]

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 3*x - exp(x)))) \\ _Michel Marcus_, Dec 02 2023

%Y Cf. A000587, A005494, A109747, A153732, A193683, A346738, A367819.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Dec 01 2023