A080417 - OEIS

%I #19 May 02 2021 21:50:46

%S 1,1,-1,1,-3,1,1,-7,6,-1,1,-15,25,-10,1,1,-31,90,-65,15,-1,1,-63,301,

%T -350,140,-21,1,1,-127,966,-1701,1050,-266,28,-1,1,-255,3025,-7770,

%U 6951,-2646,462,-36,1,1,-511,9330,-34105,42525,-22827,5880,-750,45,-1

%N Signed Stirling numbers of the second kind.

%C Define (n+1) X (n+1) matrices by M(n)=(binomial(i+1,j),i,j=0..n). The coefficients of the characteristic polynomials of these matrices yield the above sequence. Note : first 1 added to complete the triangle.

%H Russell Merris, <a href="http://journals.tubitak.gov.tr/math/issues/mat-00-24-4/mat-24-4-7-0009-1.pdf">The p-Stirling Numbers</a>, Turk J Math, 24 (2000), 379-399. See p. 3.

%F T(1, 1)=1, T(1, k)=0, k>1. T(n, k) = -T(n-1, k-1) + k * T(n, k-1), n>1.

%F abs(T(n,k)) = A008277(n,k). - _Joerg Arndt_, May 02 2021

%e Rows are

%e 1;

%e 1,  -1;

%e 1,  -3,  1;

%e 1,  -7,  6,  -1;

%e 1, -15, 25, -10, 1;

%e ...

%e 25 = -(-7) + 3*6, -10 = -6 + 4*(-1).

%o (PARI) T(n,k) = (-1)^(k+1)*stirling(n, k, 2); \\ _Michel Marcus_, May 02 2021

%Y Cf. A008277.

%K easy,sign,tabl

%O 1,5

%A _Paul Barry_, Feb 18 2003