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 A048594 Triangle a(n,k) = k! * Stirling1(n,k), 1<=k<=n. 22
 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 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: A291185 A320140 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 October 22 06:02 EDT 2018. Contains 316432 sequences. (Running on oeis4.)