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A302557 Expansion of 1/(2 - Sum_{k>=0} k!*x^k/(1 + x)^(k+1)).

%I A302557 #16 Feb 16 2025 08:33:53
%S A302557 1,0,1,2,10,48,288,1984,15660,139312,1380484,15080152,180017780,
%T A302557 2331038048,32537274756,486942025288,7777172706308,132025174277392,
%U A302557 2373753512469972,45059504242538328,900498975768121972,18898334957168597184,415537355533831049572,9552918187197519923176
%N A302557 Expansion of 1/(2 - Sum_{k>=0} k!*x^k/(1 + x)^(k+1)).
%C A302557 Invert transform of A000166.
%H A302557 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A302557 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Subfactorial.html">Subfactorial</a>
%F A302557 G.f.: 1/(1 - Sum_{k>=1} A000166(k)*x^k).
%F A302557 G.f.: 1/(2 - 1/(1 - x^2/(1 - 2*x - 4*x^2/(1 - 4*x - 9*x^2/(1 - 6*x - 16*x^2/(1 - ...)))))), a continued fraction.
%F A302557 a(n) ~ exp(-1) * n! * (1 + 2/n^2 + 6/n^3 + 35/n^4 + 256/n^5 + 2187/n^6 + 21620/n^7 + 243947/n^8 + 3098528/n^9 + 43799819/n^10 + ...), for coefficients see A305275. - _Vaclav Kotesovec_, Aug 18 2018
%t A302557 nmax = 23; CoefficientList[Series[1/(2 - Sum[k! x^k/(1 + x)^(k + 1), {k, 0, nmax}]), {x, 0, nmax}], x]
%t A302557 nmax = 23; CoefficientList[Series[1/(1 - Sum[Round[k!/Exp[1]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A302557 a[0] = 1; a[n_] := a[n] = Sum[Subfactorial[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
%Y A302557 Cf. A000166, A051295, A051296, A259869, A259870, A305275.
%K A302557 nonn
%O A302557 0,4
%A A302557 _Ilya Gutkovskiy_, Aug 15 2018