A075181 Coefficients of certain polynomials (rising powers).

1, 2, 1, 6, 6, 2, 24, 36, 22, 6, 120, 240, 210, 100, 24, 720, 1800, 2040, 1350, 548, 120, 5040, 15120, 21000, 17640, 9744, 3528, 720, 40320, 141120, 231840, 235200, 162456, 78792, 26136, 5040, 362880, 1451520, 2751840, 3265920, 2693880, 1614816

Offset: 1

Author

Wolfdieter Lang, Sep 19 2002

Keywords

• nonn
• easy
• tabl

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 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.

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

Examples

			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).
		

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.

Crossrefs

Cf. A048594, A075178, A007840, A075182.

Cf. A094638.

Programs

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 *)

PARI

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

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.