|
%I #11 Mar 26 2019 11:53:13
%S 0,1,2,5,12,1,-450,-6931,-89096,-1120895,-14394150,-191263051,
%T -2638282812,-37716883775,-556075744042,-8385570334051,
%U -127637336779920,-1916072623603199,-27033275598036174,-311878728377256475,-918069644450841860,120594465496571606401,6362190374664242284782
%N Expansion of e.g.f. sin(x/(1 - x)).
%C Lah transform of the sequence 0, 1, 0, -1, 0, 1, 0, -1, ...
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = Sum_{k=1..floor((n+1)/2)} (-1)^(k+1)*binomial(n-1,2*k-2)*n!/(2*k-1)!.
%p a:=series(sin(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[Sin[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^(k + 1) Binomial[n - 1, 2 k - 2] n!/(2 k - 1)!, {k, Floor[(n + 1)/2]}], {n, 0, 22}]
%t Join[{0}, Table[n! HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {1/2, 1, 3/2}, -1/4], {n, 22}]]
%Y Cf. A088313, A219613, A317409.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jul 27 2018
|
