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%I #56 Feb 16 2022 23:22:07
%S 1,1,-1,2,-3,1,6,-12,7,-1,24,-60,50,-15,1,120,-360,390,-180,31,-1,720,
%T -2520,3360,-2100,602,-63,1,5040,-20160,31920,-25200,10206,-1932,127,
%U -1,40320,-181440,332640,-317520,166824,-46620,6050,-255,1,362880,-1814400,3780000,-4233600,2739240,-1020600,204630,-18660,511,-1
%N Triangle of coefficients of polynomials H(n,x) formed from the first (n+1) terms of the power series expansion of ( -x/log(1-x) )^(n+1), multiplied by n!.
%C Special values: H(n,1)=0, H(2n,2)=0, H(n,-x) ~= ( x/log(1+x) )^(n+1), for x>0. H'(n,1) = -1/n!, where H'(n,x) = d/dx H(n,x).
%C The zeros of these polynomials are all positive reals >= 1. If we order the zeros of H(n,x), {r_k, k=0..(n-1)}, by magnitude so that r_0 = 1, r_k > r_(k-1), for 0 < k < n, then r_(n-k) = r_k/(r_k - 1) when 0 < k < n, n > 1, where r_(n/2) = 2 for even n.
%C Also Product_{k=0..(n-1)} r_k = n!, r_(n-1) ~ C 2^n.
%C I believe that these numbers are the coefficients of the Eulerian polynomials An(z) written in powers of z-1. That is, the sequence is: A0(1); A1(1), A1'(1); A2(1), A2'(1), A2''(1)/2!; A3(1), A3'(1), A3''(1)/2!, A3'''(1)/3!; A4(1), A4'(1), A4''(1)/2!, A4'''(1)/3!, A4''''(1)/4! etc. My convention: A0(z)=z, A1(z)=z, A2(z)=z+z^2, A3(z)=z+4z^2+z^3, A4(z)=z+11z^2+11z^3+z^4, etc. - Louis Zulli (zullil(AT)lafayette.edu), Jan 19 2005
%C H(n,2) gives 1,-1,0,2,0,-16,0,272,0,-7936,0,..., see A009006. - _Philippe Deléham_, Aug 20 2007
%C Row sums are zero except for first row. - _Roger L. Bagula_, Sep 11 2008
%C From _Groux Roland_, May 12 2011: (Start)
%C Let f(x) = (exp(x)+1)^(-1) then the n-th derivative of f equals Sum_{k=0..n} T(n,k)*(f(x))^(n+1-k).
%C T(n+1,0) = (n+1)*T(n,0); T(n+1,n+1) = -T(n,n) and for 0 < k < n T(n+1,k) = (n+1-k) * T(n,k) - (n-k+2)*T(n,k-1).
%C T(n,k) = Sum_{i=0..k} (-1)^(i+k)*(n-i)!*binomial(n-i,k-i)*S(n,n-i) where S(n,k) is a Stirling number of the second kind. (End)
%H G. C. Greubel, <a href="/A075263/b075263.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Nguyen-Huu-Bong, <a href="https://doi.org/10.1016/0097-3165(71)90049-5">Some Combinatorial Properties of Summation Operators</a>, J. Comb. Theory, Ser. A 11.3 (1971): 213-221. See Table on page 214.
%F Generated by [1, 1, 2, 2, 3, 3, ...] DELTA [ -1, 0, -2, 0, -3, 0, ...], where DELTA is the operator defined in A084938.
%F T(n, k) = Sum_{i=0..n-k} (-1)^(n-i)*C(n-k, i)*(i+1)^n; n >= 0, 0 <= k <= n. - _Paul D. Hanna_, Jul 21 2005
%F E.g.f.: A(x, y) = -log(1-(1-exp(-x*y))/y). - _Paul D. Hanna_, Jul 21 2005
%F p(x,n) = x^(n + 1)*Sum_{k>=0} k^n*(1 - x)^k; t(n,m) = Coefficients(p(x,n)). - _Roger L. Bagula_, Sep 11 2008
%F p(x,n) = x^(n + 1)*PolyLog(-n, 1 - x); t(n,m) = coefficients(p(x,n)) for n >= 1. - _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008
%e H(0,x) = 1
%e H(1,x) = (1 - 1*x)/1!
%e H(2,x) = (2 - 3*x + 1*x^2)/2!
%e H(3,x) = (6 - 12*x + 7*x^2 - 1*x^3)/3!
%e H(4,x) = (24 - 60*x + 50*x^2 - 15*x^3 + 1*x^4)/4!
%e H(5,x) = (120 - 360*x + 390*x^2 - 180*x^3 + 31*x^4 - 1*x^5)/5!
%e H(6,x) = (720 - 2520*x + 3360*x^2 - 2100*x^3 + 602*x^4 - 63*x^5 + 1*x^5)/6!
%e Triangle begins:
%e 1;
%e 1, -1;
%e 2, -3, 1;
%e 6, -12, 7, -1;
%e 24, -60, 50, -15, 1;
%e 120, -360, 390, -180, 31, -1;
%e 720, -2520, 3360, -2100, 602, -63, 1;
%e 5040, -20160, 31920, -25200, 10206, -1932, 127, -1;
%p CL := f -> PolynomialTools:-CoefficientList(f,x):
%p T_row := n -> `if`(n=0, [1], CL(x^(n+1)*polylog(-n, 1-x))):
%p for n from 0 to 6 do T_row(n) od; # _Peter Luschny_, Sep 28 2017
%t Table[CoefficientList[x^(n+1)*Sum[k^n*(1-x)^k, {k, 0, Infinity}], x], {n, 0, 10}]//Flatten (* _Roger L. Bagula_, Sep 11 2008 *)
%t p[x_, n_]:= x^(n+1)*PolyLog[-n, 1-x]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten (* _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008 *)
%o (PARI) T(n,k)=if(k<0 || k>n,0,n!*polcoeff((-x/log(1-x+x^2*O(x^n)))^(n+1),k))
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (PARI) T(n,k)=sum(i=0,n-k,(-1)^(n-i)*binomial(n-k,i)*(i+1)^n)
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (PARI) /* Using e.g.f. A(x,y): */
%o {T(n,k)=local(X=x+x*O(x^n),Y=y+y^2*O(y^(k))); n!*polcoeff(polcoeff(-log(1-(1-exp(-X*Y))/y),n,x),k,y)}
%o for(n=0,10,for(k=0,n-1,print1(T(n,k),", "));print(""))
%o (PARI) /* Deléham's DELTA: T(n,k) = [x^(n-k)*y^k] P(n,0) */
%o {P(n,k)=if(n<0||k<0,0,if(n==0,1, P(n,k-1)+(x*(k\2+1)+y*(-(k\2+1)*((k+1)%2)))*P(n-1,k+1)))}
%o {T(n,k)=polcoeff(polcoeff(P(n,0),n-k,x),k,y)}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (Magma)
%o T:= func< n,k | &+[(-1)^(n-j)*Binomial(n-k,j)*(j+1)^n: j in [0..n-k]] >;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 27 2020
%o (Sage)
%o def T(n, k): return sum( (-1)^(n-j)*binomial(n-k, j)*(j+1)^n for j in (0..n-k))
%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 27 2020
%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..n-k], j->
%o (-1)^(n-j)*Binomial(n-k,j)*(j+1)^n )))); # _G. C. Greubel_, Jan 27 2020
%Y Cf. A028246, A075264, A084938, A123125.
%Y Cf. Eulerian numbers (A008292).
%K nice,sign,tabl
%O 0,4
%A _Paul D. Hanna_, Sep 13 2002
%E Error in one term corrected by _Benoit Cloitre_, Aug 20 2007
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