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A048594 Triangle a(n,k) = k! * Stirling1(n,k), 1<=k<=n. 19
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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).

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

FORMULA

a(n, k) = k*a(n-1, k-1)-(n-1)*a(n-1, k) if n>=k>=1, a(n, 0) := 0 and a(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)

EXAMPLE

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.

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 *)

PROG

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

(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

CROSSREFS

Cf. A008275, A019538, A075181.

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

Sequence in context: A122766 A291185 A033742 * A178801 A130493 A267516

Adjacent sequences:  A048591 A048592 A048593 * A048595 A048596 A048597

KEYWORD

sign,tabl,easy,nice,look

AUTHOR

Oleg Marichev (oleg(AT)wolfram.com)

STATUS

approved

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Last modified February 21 21:41 EST 2018. Contains 299427 sequences. (Running on oeis4.)