OFFSET
1,2
COMMENTS
This is the unsigned triangle A048594 with rows read backwards.
The row polynomials p(n,y) := Sum_{m=0..n-1}a(n,m)*y^m, n>=1, are obtained from (log(x)*(-x*log(x))^n)*(d^n/dx^n)(1/log(x)), n>=1, after replacement of log(x) by y.
The columns give A000142 (factorials), A001286 (Lah), 2* A075183, 2*A075184, 4*A075185, 4!*A075186, 4!*A075187 for m=0..6.
Coefficients T(n,k) of the differential operator expansion
[x^(1+y)D]^n = x^(n*y)[T(n,1)* (xD)^n / n! + y * T(n,2)* (xD)^(n-1) / (n-1)! + ... + y^(n-1) * T(n,n) * (xD)], where D = d/dx. Note that (xD)^n = Bell(n,:xD:), where (:xD:)^n = x^n * D^n and Bell(n,x) are the Bell / Touchard polynomials. See A094638. - Tom Copeland, Aug 22 2015
LINKS
Vincenzo Librandi, Rows n = 1..100, flattened
Y.-Z. Huang, J. Lepowsky and L. Zhang, A logarithmic generalization of tensor product theory for modules for a vertex operator algebra, arXiv:math/0311235 [math.QA], 2003; 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. Equal Sums of Reciprocal Products: 10992 (2004) pp. 827-829.
FORMULA
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.
EXAMPLE
Triangle starts:
1;
2,1;
6,6,2;
24,36,22,6;
...
n=2: (x^2*log(x)^3)*(d^2/d^x^2)(1/log(x)) = 2 + log(x).
MAPLE
seq(seq(k!*abs(Stirling1(n, k)), k=n..1, -1), n=1..10); # Robert Israel, Jul 12 2015
MATHEMATICA
Table[ Table[ k!*StirlingS1[n, k] // Abs, {k, 1, n}] // Reverse, {n, 1, 9}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
PROG
(PARI) {T(n, k)= if(k<0 || k>=n, 0, (-1)^k* stirling(n, n-k)* (n-k)!)} /* Michael Somos Apr 11 2007 */
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 19 2002
STATUS
approved