A080417 Signed Stirling numbers of the second kind.

1, 1, -1, 1, -3, 1, 1, -7, 6, -1, 1, -15, 25, -10, 1, 1, -31, 90, -65, 15, -1, 1, -63, 301, -350, 140, -21, 1, 1, -127, 966, -1701, 1050, -266, 28, -1, 1, -255, 3025, -7770, 6951, -2646, 462, -36, 1, 1, -511, 9330, -34105, 42525, -22827, 5880, -750, 45, -1

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..55.

Russell Merris, The p-Stirling Numbers, Turk J Math, 24 (2000), 379-399. See p. 3.

Formula

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.

abs(T(n,k)) = A008277(n,k). - Joerg Arndt, May 02 2021

Example

Rows are
1;
1,  -1;
1,  -3,  1;
1,  -7,  6,  -1;
1, -15, 25, -10, 1;
...
25 = -(-7) + 3*6, -10 = -6 + 4*(-1).

Prog

(PARI) T(n, k) = (-1)^(k+1)*stirling(n, k, 2); \\ Michel Marcus, May 02 2021

Crossrefs

Cf. A008277.

Sequence in context: A250118 A250119 A154959 * A008277 A218577 A193387

Adjacent sequences: A080414 A080415 A080416 * A080418 A080419 A080420

Keyword

easy,sign,tabl

Author

Paul Barry, Feb 18 2003

Status

approved