1,2

Sum {k=1..n} C(n,k)*(n-1)!/(k-1)! = A000262(n) = 1,3,13,73,501,4051,37633,394353,4596553,58941091,...

n*Sum {k=1..n}C(n,k)*(n-1)!/(k-1)! = A103194 = 1,6,39,292,2505,24306,263431,3154824,...

Sum {k=1..n}C(n,k)* (n-1)!/(k-1)!*k^2 = A103194 = 1,6,39,292,2505,24306,263431,3154824,...

n*Sum C(n,k)*(n-1)!/(k-1)! = Sum {k=1..n}C(n,k)*(n-1)!/(k-1)!*k^2

T(n,k) is the number of partially ordered sets (posets) on n elements that consist entirely of k chains. For example, T(4, 3)=12 since there are exactly 12 of posets on {a,b,c,d}that consist entirely of 3 chains. Letting ab denote a<=b and using a slash "/" to separate chains, the 12 posets can be given by a/b/cd, a/b/dc, a/c/bd, a/c/db, a/d/bc, a/d/cb, b/c/ad, b/c/da, b/d/ac, b/d/ca, c/d/ab, and c/d/ba, where the listing of the chains is arbitrary (e.g., a/b/cd = a/cd/b =...cd/b/a). - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Feb 22 2007

Also the matrix product |S1|.S2 of Stirling numbers of both kinds.

This Lah triangle is a lower triangular matrix of the Jabotinsky type. See the column e.g.f. and the D. E. Knuth reference given in A008297. - W. Lang, Jun 29 2007

The infinitesimal matrix generator of this matrix is given in A132710. See A111596 for an interpretation in terms of circular binary words and generalized factorials. - Tom Copeland (tcjpn(AT)msn.com), Nov 22 2007

MacTutor History of Mathematics archive: Biography of Ivo Lah.

T(n,k) = sum(|S1(n,m)|*S2(m,k),m=n..k), k>= n>=1, with Stirling triangles S2(n,m):=A048993 and S1(n,m):=A048994.

T(n,k) = C(n,k)*(n-1)!/(k-1)!.

E.g.f. column k: (x^(k-1)/(1-x)^(k+1))/(k-1)!, k>=1.

Recurrence from Sheffer (here Jabotinsky) a-sequence [1,1,0,...] (see the W. Lang link under A006232): T(n,k)=(n/k)*T(n-1,m-1) + n*T(n-1,m). W. Lang, Jun 29 2007

The e.g.f. is, umbrally, exp[(.)!* L(.,-t,1)*x] = exp[t*x/(1-x)]/(1-x)^2 where L(n,t,1) = sum(k=0,...,n) T(n+1,k+1)*(-t)^k = sum(k=0,...,n) binomial(n+1,k+1)* (-t)^k / k! is the associated Laguerre polynomial of order 1. - Tom Copeland (tcjpn(AT)msn.com), Nov 17 2007

T(1,1) = C(1,1)*0!/0! = 1,

T(2,1) = C(2,1)*1!/0! = 2,

T(2,2) = C(2,2)*1!/1! = 1,

T(3,1) = C(3,1)*2!/0! = 6,

T(3,2) = C(3,2)*2!/1! = 6,

T(3,3) = C(3,3)*2!/2! = 1,

Sheffer a-sequence recurrence: T(6,2)= 1800 = (6/3)*120 +6*240.

The triangle: for n from 1 to 13 do seq(binomial(n, k)*(n-1)!/(k-1)!, k=1..n) od; the sequence: seq(seq(binomial(n, k)*(n-1)!/(k-1)!, k=1..n), n=1..13);

Cf. A000262, A103194, A105220.

Triangle of Lah numbers (A008297) unsigned.

Cf. |A111596(n, m)| (triangle with extra n=0 row and m=0 column).

Sequence in context: A130561 A091599 A066667 this_sequence A008297 A048999 A090582

Adjacent sequences: A105275 A105276 A105277 this_sequence A105279 A105280 A105281

easy,nonn,tabl

Miklos Kristof (kristmikl(AT)freemail.hu), Apr 25 2005

Stirling comments and e.g.f.s from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Apr 11 2007.