0,8

The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.

Mirror image of the triangle A054654 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 30 2006

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.

L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 245.

J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

R. M. Dickau, Stirling numbers of the first kind

s(n, k)=s(n-1, k-1)-(n-1)*s(n-1, k), n, k >= 1; s(n, 0)=s(0, k)=0; s(0, 0)=1.

The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0)=a(0, k)=0; a(0, 0)=1.

Triangle (signed) = [0, -1, -1, -2, -2, -3, -3, -4, -4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; Triangle(unsigned) = [0, 1, 1, 2, 2, 3, 3, 4, 4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deleham's operator defined in A084938.

Sum_{k=0..n} (-m)^(n-k)*T(n, k) = A000142(n), A001147(n), A007559(n), A007696(n), ... for m = 1, 2, 3, 4, ... .- Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2005

A008275*A007318 as infinite lower triangular matrices. [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Aug 20 2009]

1; 0,1; 0,-1,1; 0,2,-3,1; 0,-6,11,-6,1; 0,24,-50,35,-10,1; ...

A048994 := proc(n, k) combinat[stirling1](n, k) ; end: - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 23 2009

seq(print(seq(coeff(expand(k!*binomial(x, k)), x, i), i=0..k)), k=0..9); [From Peter Luschny (peter(AT)luschny.de), Jul 13 2009]

(PARI) a(n, k) = if(k<0|k>n, 0, if(n==0, 1, (n-1)*a(n-1, k)+a(n-1, k-1)))

See especially A008275 which is the main entry for this triangle. Cf. A008277, A039814-A039817, A048993.

Cf. A084938.

A000142(n) = sum(|s(n, k)|) k=0..n, n >= 0.

Sequence in context: A081247 A144633 A005210 this_sequence A132393 A121434 A137329

Adjacent sequences: A048991 A048992 A048993 this_sequence A048995 A048996 A048997

sign,tabl,nice

N. J. A. Sloane (njas(AT)research.att.com).

Offset corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 23 2009

Formula corrected by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 10 2009