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A080417
Signed Stirling numbers of the second kind.
0
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
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
KEYWORD
AUTHOR
Paul Barry, Feb 18 2003
STATUS
approved