A103879 - OEIS

%I #6 Apr 30 2014 01:31:16

%S 1,1,0,1,-1,0,1,-3,1,0,1,-11,7,-1,0,1,-25,85,-15,1,0,1,-137,415,-575,

%T 31,-1,0,1,-49,12019,-5845,3661,-63,1,0,1,-121,13489,-874853,76111,

%U -22631,127,-1,0,1,-761,726301,-336581,58067611,-952525,137845,-255,1

%N Square array T(n,k) read by antidiagonals: numerators of Stirling numbers of first kind with negative argument S1(-n,k), n,k>=0.

%H D. Loeb, <a href="http://arXiv.org/abs/math.CO/9502217">Generalization of the Stirling numbers of first kind</a>

%F T(n, k) = (-1)^(k+1) * Sum[i=1..n, C(n, i)*(-1)^i*i^(-k) ].

%F G.f. of n-th row: 1/n! * 1/Prod[i=1..n, 1+x/i ].

%e 1, 0, 0, 0, 0, 0,

%e 1, -1, 1, -1, 1, -1,

%e 1/2, -3/4, 7/8, -15/16, 31/32, -63/64,

%e 1/6, -11/36, 85/216, -575/1296, 3661/7776, -22631/46656,

%e 1/24,-25/288,415/3456,-5845/41472,76111/497664,-952525/5971968,

%o (PARI) T(n,k)=numerator(1/n!*polcoeff(Ser(1/prod(i=1,n,1+x/i)),k))

%Y Denominators are in A103880. Cf. A008969.

%K sign,tabl,frac

%O 0,8

%A _Ralf Stephan_, Feb 20 2005