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%I #15 Mar 26 2019 16:15:37
%S 1,0,-1,-6,-35,-220,-1501,-10962,-83495,-632952,-4260601,-13852190,
%T 355180981,12991115436,320077652075,7153866992790,155785273182001,
%U 3395838000334352,75000970329466895,1687941779356532682,38803334491247820301,911633573138881234740,21870615120012355726259
%N Expansion of e.g.f. cos(x/(1 - x)).
%C Lah transform of the sequence 1, 0, -1, 0, 1, 0, -1, 0, ...
%H Robert Israel, <a href="/A317409/b317409.txt">Table of n, a(n) for n = 0..446</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-1,2*k-1)*n!/(2*k)!.
%F -2*(2*n + 3)*(n + 2)*(n + 1)*a(n + 1) + (6*n^2 + 24*n + 25)*a(n + 2) - 2*(2*n + 5)*a(n + 3) + a(n + 4) + n*(n + 2)*(n + 1)^2*a(n)=0. - _Robert Israel_, Mar 26 2019
%p a:=series(cos(x/(1 - x)), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # _Paolo P. Lava_, Mar 26 2019
%t nmax = 22; CoefficientList[Series[Cos[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^k Binomial[n - 1, 2 k - 1] n!/(2 k)!, {k, 0, Floor[n/2]}], {n, 0, 22}]
%t Join[{1}, Table[(1 - n) n! HypergeometricPFQ[{1 - n/2, 3/2 - n/2}, {3/2, 3/2, 2}, -1/4]/2, {n, 22}]]
%o (PARI) my(x='x + O('x^25)); Vec(serlaplace(cos(x/(1 - x)))) \\ _Michel Marcus_, Mar 26 2019
%Y Cf. A056594, A088312, A219613, A317406, A317410.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Jul 27 2018
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