Original entry on oeis.org
2, 0, 8, 96, 1248, 18176, 295040, 5294592, 104269312, 2239225856, 52150118400, 1310675066880, 35390943453184, 1022570290544640, 31498715705147392, 1030904079324413952, 35736902010299351040, 1308417934560279396352
Offset: 1
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terms = 18; f[t_] = 1 + Sum[n! t^n, {n, 1, terms+1}];
A122827 = CoefficientList[(f[t] - 1)/f[t]^2 + O[t]^(terms+1), t] // Rest;
2^Range[terms]*A122827 (* Jean-François Alcover, Feb 13 2019 *)
A355488
Expansion of g.f. f/(1+2*f) where f is the g.f. of nonempty permutations.
Original entry on oeis.org
0, 1, 0, 2, 8, 48, 328, 2560, 22368, 216224, 2291456, 26430336, 329805952, 4429255168, 63730438656, 978479250944, 15972310317056, 276292865550336, 5049672714569728, 97245533647568896, 1968395389124714496, 41783552069858877440, 928204423021249003520
Offset: 0
Consider the permutations of [3]: [2,3,1], [3,1,2] and [3,2,1] have 1 component,
[1,3,2] and [2,1,3] have 2 components, and [1,2,3] has three components. Thus 3 - 2 + 1 = 2 = a(3). - _Peter Luschny_, Sep 10 2022
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a:= n-> (f-> coeff(series(f/(1+2*f), x, n+1), x, n))(add(j!*x^j, j=1..n)):
seq(a(n), n=0..23); # Alois P. Heinz, Jul 20 2022
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nmax=22; f[x_]:=Sum[i! x^i,{i,nmax}]; CoefficientList[Series[f[x]/(1+2f[x]),{x,0,nmax}],x] (* Stefano Spezia, Jul 04 2022 *)
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A = QQ[['t']]
f = A([0] + [factorial(n) for n in range(1,30)]).O(30)
print(list(f/(1+2*f)))
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# Uses function A059438_triangle.
def A355488_list(size):
triangle = A059438_triangle(size)
return [0] + [sum((-1)^k*t for (k,t) in enumerate(row)) for row in triangle]
print(A355488_list(20)) # Peter Luschny, Sep 10 2022
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