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 A075181 Coefficients of certain polynomials (rising powers). 10
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 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, 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. 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. EXAMPLE 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). 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 *) PROG (PARI) {T(n, k)= if(k<0| k>=n, 0, (-1)^k* stirling(n, n-k)* (n-k)!)} /* Michael Somos Apr 11 2007 */ CROSSREFS Cf. A048594, A075178, A007840, A075182. Cf. A094638. Sequence in context: A271881 A182729 A260885 * A052121 A193895 A193561 Adjacent sequences:  A075178 A075179 A075180 * A075182 A075183 A075184 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Sep 19 2002 STATUS approved

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Last modified March 23 10:55 EDT 2019. Contains 321424 sequences. (Running on oeis4.)