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

1, 0, 1, 2, 10, 48, 288, 1984, 15660, 139312, 1380484, 15080152, 180017780, 2331038048, 32537274756, 486942025288, 7777172706308, 132025174277392, 2373753512469972, 45059504242538328, 900498975768121972, 18898334957168597184, 415537355533831049572, 9552918187197519923176

Offset

0,4

Comments

Invert transform of A000166.

Links

Table of n, a(n) for n=0..23.

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Subfactorial

Formula

G.f.: 1/(1 - Sum_{k>=1} A000166(k)*x^k).

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.

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

Mathematica

nmax = 23; CoefficientList[Series[1/(2 - Sum[k! x^k/(1 + x)^(k + 1), {k, 0, nmax}]), {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 - Sum[Round[k!/Exp[1]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Subfactorial[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

Crossrefs

Cf. A000166, A051295, A051296, A259869, A259870, A305275.

Sequence in context: A200540 A378907 A166922 * A129118 A037256 A270664

Adjacent sequences: A302554 A302555 A302556 * A302558 A302559 A302560

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 15 2018

Status

approved