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

%I A048594 #53 Feb 16 2025 08:32:40
%S A048594 1,-1,2,2,-6,6,-6,22,-36,24,24,-100,210,-240,120,-120,548,-1350,2040,
%T A048594 -1800,720,720,-3528,9744,-17640,21000,-15120,5040,-5040,26136,-78792,
%U A048594 162456,-235200,231840,-141120,40320,40320,-219168,708744,-1614816,2693880,-3265920,2751840,-1451520,362880
%N A048594 Triangle T(n,k) = k! * Stirling1(n,k), 1<=k<=n.
%C A048594 Row sums (unsigned) give A007840(n), n>=1; (signed): A006252(n), n>=1.
%C A048594 Apart from signs, coefficients in expansion of n-th derivative of 1/log(x).
%H A048594 Reinhard Zumkeller, <a href="/A048594/b048594.txt">Rows n = 1..125 of triangle, flattened</a>
%H A048594 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Stirling Number of the First Kind</a>
%H A048594 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_numbers_and_exponential_generating_functions">Stirling numbers and exponential generating functions</a>
%F A048594 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.
%F A048594 E.g.f. k-th column: log(1+x)^k, k>=1.
%F A048594 From _Peter Bala_, Nov 25 2011: (Start):
%F A048594 E.g.f.: 1/(1-t*log(1+x)) = 1 + t*x + (-t+2*t^2)*x^2/2! + ....
%F A048594 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.
%F A048594 (End)
%e A048594 Triangle begins
%e A048594    1;
%e A048594   -1,    2;
%e A048594    2,   -6,   6;
%e A048594   -6,   22, -36,   24;
%e A048594   24, -100, 210, -240, 120; ...
%e A048594 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.
%p A048594 with(combinat): A048594 := (n,k)->k!*stirling1(n,k);
%t A048594 Flatten[Table[k!*StirlingS1[n,k], {n,10}, {k,n}]] (* _Harvey P. Dale_, Aug 28 2011 *)
%t A048594 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 *)
%o A048594 (PARI) {T(n, k)= if(k<1 || k>n, 0, stirling(n, k)* k!)} /* _Michael Somos_ Apr 11 2007 */
%o A048594 (Haskell)
%o A048594 a048594 n k = a048594_tabl !! (n-1) !! (k-1)
%o A048594 a048594_row n = a048594_tabl !! (n-1)
%o A048594 a048594_tabl = map snd $ iterate f (1, [1]) where
%o A048594    f (i, xs) = (i + 1, zipWith (-) (zipWith (*) [1..] ([0] ++ xs))
%o A048594                                    (map (* i) (xs ++ [0])))
%o A048594 -- _Reinhard Zumkeller_, Mar 02 2014
%o A048594 (Magma) /* As triangle: */ [[Factorial(k)*StirlingFirst(n,k): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Dec 15 2015
%o A048594 (SageMath)
%o A048594 def A048594(n,k): return (-1)^(n-k)*factorial(k)*stirling_number1(n,k)
%o A048594 flatten([[A048594(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Oct 24 2023
%Y A048594 Cf. A008275, A019538, A075181.
%Y A048594 Cf. A133942 (left edge), A000142 (right edge), A006252 (row sums), A238685 (central terms).
%Y A048594 Row sums: A007840 (unsigned), A006252 (signed).
%K A048594 sign,tabl,easy,nice,look
%O A048594 1,3
%A A048594 Oleg Marichev (oleg(AT)wolfram.com)