A048594 Triangle T(n,k) = k! * Stirling1(n,k), 1<=k<=n.

1, -1, 2, 2, -6, 6, -6, 22, -36, 24, 24, -100, 210, -240, 120, -120, 548, -1350, 2040, -1800, 720, 720, -3528, 9744, -17640, 21000, -15120, 5040, -5040, 26136, -78792, 162456, -235200, 231840, -141120, 40320, 40320, -219168, 708744, -1614816, 2693880, -3265920, 2751840, -1451520, 362880

Offset: 1

Author

Oleg Marichev (oleg(AT)wolfram.com)

Keywords

• sign
• tabl
• easy
• nice
• look

Comments

Row sums (unsigned) give A007840(n), n>=1; (signed): A006252(n), n>=1.

Apart from signs, coefficients in expansion of n-th derivative of 1/log(x).

Examples

			Triangle begins
   1;
  -1,    2;
   2,   -6,   6;
  -6,   22, -36,   24;
  24, -100, 210, -240, 120; ...
The 2nd derivative of 1/log(x) is -2/x^3*log(x)^2 - 6/x^3*log(x)^3 - 6/x^3*log(x)^4.
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
• Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
• Wikipedia, Stirling numbers and exponential generating functions

Crossrefs

Cf. A008275, A019538, A075181.

Cf. A133942 (left edge), A000142 (right edge), A006252 (row sums), A238685 (central terms).

Row sums: A007840 (unsigned), A006252 (signed).

Programs

Haskell

a048594 n k = a048594_tabl !! (n-1) !! (k-1)
a048594_row n = a048594_tabl !! (n-1)
a048594_tabl = map snd $ iterate f (1, [1]) where
   f (i, xs) = (i + 1, zipWith (-) (zipWith (*) [1..] ([0] ++ xs))
                                   (map (* i) (xs ++ [0])))
-- Reinhard Zumkeller, Mar 02 2014

Magma

/* As triangle: */ [[Factorial(k)*StirlingFirst(n,k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Dec 15 2015

Maple

with(combinat): A048594 := (n,k)->k!*stirling1(n,k);

Mathematica

Flatten[Table[k!*StirlingS1[n,k], {n,10}, {k,n}]] (* Harvey P. Dale, Aug 28 2011 *)
Join @@ CoefficientRules[ -Table[ D[ 1/Log[z], {z, n}], {n, 9}] /. Log[z] -> -Log[z], {1/z, 1/Log[z]}, "NegativeLexicographic"][[All, All, 2]] (* Oleg Marichev (oleg(AT)wolfram.com) and Maxim Rytin (m.r(AT)inbox.ru); submitted by Robert G. Wilson v, Aug 29 2011 *)

PARI

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

SageMath

def A048594(n,k): return (-1)^(n-k)*factorial(k)*stirling_number1(n,k)
flatten([[A048594(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 24 2023

Formula

T(n, k) = k*T(n-1, k-1) - (n-1)*T(n-1, k) if n>=k>=1, T(n, 0) = 0 and T(1, 1)=1, else 0.

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

From Peter Bala, Nov 25 2011: (Start):

E.g.f.: 1/(1-t*log(1+x)) = 1 + t*x + (-t+2*t^2)*x^2/2! + ....

The row polynomials are given by D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator exp(-x)*d/dx.

(End)