0,4

Special values: H(n,1)=0, H(2n,2)=0, H(n,-x) ~= ( x/ln(1+x) )^(n+1), for x>0. H'(n,1) = -1/n!, where H'(n,x) = d/dx H(n,x).

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.

Also product_{k=0..(n-1)} r_k = n!, r_(n-1) ~ C 2^n.

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

H(n,2) gives 1,-1,0,2,0,-16,0,272,0,-7936,0,..., see A009006 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 20 2007

Generated by [1, 1, 2, 2, 3, 3, ...] DELTA [ -1, 0, -2, 0, -3, 0, ...], where DELTA is the operator defined in A084938.

T(n, k) = Sum_{i=0..n-k} (-1)^(n-i)*C(n-k, i)*(i+1)^n; n >= 0, 0<=k<=n. E.g.f.: A(x, y) = -log(1-(1-exp(-x*y))/y). - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 21 2005

H(0,x) = 1

H(1,x) = (1 - 1x)/1!

H(2,x) = (2 - 3x + 1x^2)/2!

H(3,x) = (6 - 12x + 7x^2 - 1x^3)/3!

H(4,x) = (24 - 60x + 50x^2 - 15x^3 + 1x^4)/4!

H(5,x) = (120 - 360x + 390x^2 - 180x^3 + 31x^4 - 1x^5)/5!

H(6,x) = (720 - 2520x + 3360x^2 - 2100x^3 + 602x^4 - 63x^5 + 1x^5)/6!

(PARI) T(n, k)=if(k<0|k>n, 0, n!*polcoeff((-x/log(1-x+x^2*O(x^n)))^(n+1), k))

(PARI) T(n, k)=sum(i=0, n-k, (-1)^(n-i)*binomial(n-k, i)*(i+1)^n) (PARI) /* Using E.g.f. A(x, y): */ {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)}

(PARI) /* DELEHAM's DELTA: T(n, k) = [x^(n-k)*y^k] P(n, 0) */ {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)))} {T(n, k)=polcoeff(polcoeff(P(n, 0), n-k, x), k, y)}

Cf. A075264, A028246, A084938.

Sequence in context: A105613 A135894 A130850 this_sequence A130405 A058372 A128264

Adjacent sequences: A075260 A075261 A075262 this_sequence A075264 A075265 A075266

nice,sign,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 13 2002

Error in one term corrected by Benoit Cloitre, Aug 20 2007