1,2

This is the unsigned triangle A048594 with rows read backwards.

The row polynomials p(n,y) := sum(a(n,m)*y^m,m=0..n-1), n>=1, are obtained from (ln(x)*(-x*ln(x))^n)*diff(1/ln(x),x$n), n>=1, after replacement of ln(x) by y. Here diff(f(x),x$n) denotes n-fold differentiation of f(x) with respect to x, n>=1.

The gcd of row n is A075182(n). Row sums give A007840(n), n>=1.

The columns give A000142 (factorials), A001286 (Lah), 2* A075183, 2*A075184, 4*A075185, 4!*A075186, 4!*A075187 for m=0..6.

Y.-Z. Huang, J. Lepowsky and L. Zhang, A logarithmic generalization of tensor product theory for modules for a vertex operator algebra, Internat. J. Math. 17 (2006), no. 8, 975--1012. see page 984 eq. (3.9) MR2261644.

D. Lubell, Problem 10992, problems and solutions, Amer. Math. Monthly 110 (2003) p. 155. Solution 111 (2004) pp. 827-829.

a(n, m)= (n-m)!*|S1(n, n-m)|, n>=m+1>=1, else 0, with S1(n, m) := A008275(n, m) (Stirling1).

a(n, m)=(n-m)*a(n-1, m)+(n-1)*a(n-1, m-1), if n>=m+1>=1, a(n, -1) := 0 and a(1, 0)=1, else 0.

1;2,1;6,6,2;24,36,22,6;...

n=2: (x^2*ln(x)^3)*diff(1/ln(x),x$2)=2+ln(x).

(PARI) {T(n, k)= if(k<0| k>=n, 0, (-1)^k* stirling(n, n-k)* (n-k)!)} /* Michael Somos Apr 11 2007 */

Cf. A048594, A075178, A007840, A075182.

Sequence in context: A048999 A090582 A079641 this_sequence A052121 A117965 A111646

Adjacent sequences: A075178 A075179 A075180 this_sequence A075182 A075183 A075184

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 19, 2002