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A075181 Coefficients of certain polynomials (rising powers).

%I A075181 #34 Jul 18 2024 21:02:11
%S A075181 1,2,1,6,6,2,24,36,22,6,120,240,210,100,24,720,1800,2040,1350,548,120,
%T A075181 5040,15120,21000,17640,9744,3528,720,40320,141120,231840,235200,
%U A075181 162456,78792,26136,5040,362880,1451520,2751840,3265920,2693880,1614816
%N A075181 Coefficients of certain polynomials (rising powers).
%C A075181 This is the unsigned triangle A048594 with rows read backwards.
%C A075181 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.
%C A075181 The gcd of row n is A075182(n). Row sums give A007840(n), n>=1.
%C A075181 The columns give A000142 (factorials), A001286 (Lah), 2* A075183, 2*A075184, 4*A075185, 4!*A075186, 4!*A075187 for m=0..6.
%C A075181 Coefficients T(n,k) of the differential operator expansion
%C A075181 [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
%H A075181 Vincenzo Librandi, <a href="/A075181/b075181.txt">Rows n = 1..100, flattened</a>
%H A075181 Y.-Z. Huang, J. Lepowsky and L. Zhang, <a href="https://arxiv.org/abs/math/0311235">A logarithmic generalization of tensor product theory for modules for a vertex operator algebra</a>, arXiv:math/0311235 [math.QA], 2003; Internat. J. Math. 17 (2006), no. 8, 975-1012. See page 984 eq. (3.9) MR2261644.
%H A075181 D. Lubell, <a href="http://www.jstor.org/stable/3647779">Problem 10992, problems and solutions</a>, Amer. Math. Monthly 110 (2003) p. 155. <a href="http://www.jstor.org/stable/4145207">Equal Sums of Reciprocal Products: 10992</a> (2004) pp. 827-829.
%F A075181 a(n, m) = (n-m)!*|S1(n, n-m)|, n>=m+1>=1, else 0, with S1(n, m) := A008275(n, m) (Stirling1).
%F A075181 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.
%e A075181 Triangle starts:
%e A075181 1;
%e A075181 2,1;
%e A075181 6,6,2;
%e A075181 24,36,22,6;
%e A075181 ...
%e A075181 n=2: (x^2*log(x)^3)*(d^2/d^x^2)(1/log(x)) = 2 + log(x).
%p A075181 seq(seq(k!*abs(Stirling1(n,k)),k=n..1,-1),n=1..10); # _Robert Israel_, Jul 12 2015
%t A075181 Table[ Table[ k!*StirlingS1[n, k] // Abs, {k, 1, n}] // Reverse, {n, 1, 9}] // Flatten (* _Jean-François Alcover_, Jun 21 2013 *)
%o A075181 (PARI) {T(n, k)= if(k<0 || k>=n, 0, (-1)^k* stirling(n, n-k)* (n-k)!)} /* _Michael Somos_ Apr 11 2007 */
%Y A075181 Cf. A048594, A075178, A007840, A075182.
%Y A075181 Cf. A094638.
%K A075181 nonn,easy,tabl
%O A075181 1,2
%A A075181 _Wolfdieter Lang_, Sep 19 2002