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A075263 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!. 6
1, 1, -1, 2, -3, 1, 6, -12, 7, -1, 24, -60, 50, -15, 1, 120, -360, 390, -180, 31, -1, 720, -2520, 3360, -2100, 602, -63, 1, 5040, -20160, 31920, -25200, 10206, -1932, 127, -1, 40320, -181440, 332640, -317520, 166824, -46620, 6050, -255, 1, 362880, -1814400, 3780000, -4233600, 2739240, -1020600 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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).

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 Deléham, Aug 20 2007

Row sums are zero except for first row. - Roger L. Bagula, Sep 11 2008

From Groux Roland, May 12 2011: (Start)

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)).

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+2-k)*T(n,k-1).

T(n,k) = sum((-1)^(i+k)*(n-i)!*binomial(n-i,k-i)*S(n,n-i),i=0..k) where S(n,k) is a Stirling number of the second kind. (End)

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

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. - Paul D. Hanna, Jul 21 2005

E.g.f.: A(x, y) = -log(1-(1-exp(-x*y))/y). - Paul D. Hanna, Jul 21 2005

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

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

EXAMPLE

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!

Triangle begins:

{1},

{1, -1},

{2, -3, 1},

{6, -12,7, -1},

{24, -60, 50, -15, 1},

{120, -360, 390, -180, 31, -1},

{720, -2520, 3360, -2100, 602, -63, 1},

{5040, -20160, 31920, -25200, 10206, -1932, 127, -1},

{40320, -181440,332640, -317520, 166824, -46620, 6050, -255, 1},

{362880, -1814400, 3780000, -4233600, 2739240, -1020600, 204630, -18660, 511,-1},

{3628800, -19958400, 46569600, -59875200, 46070640, -21538440, 5921520, -874500,57002, -1023, 1}

...

MAPLE

CL := f -> PolynomialTools:-CoefficientList(f, x):

T_row := n -> `if`(n=0, [1], CL(x^(n+1)*polylog(-n, 1-x))):

for n from 0 to 6 do T_row(n) od; # Peter Luschny, Sep 28 2017

MATHEMATICA

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 *)

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 *)

PROG

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

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

(PARI) T(n, k)=sum(i=0, n-k, (-1)^(n-i)*binomial(n-k, i)*(i+1)^n)

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

(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)}

for(n=0, 10, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

(PARI) /* Deléham'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)}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A075264, A028246, A084938, A123125.

Cf. Eulerian numbers (A008292).

Sequence in context: A263634 A135894 A247500 * A130850 A130405 A058372

Adjacent sequences:  A075260 A075261 A075262 * A075264 A075265 A075266

KEYWORD

nice,sign,tabl

AUTHOR

Paul D. Hanna, Sep 13 2002

EXTENSIONS

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

STATUS

approved

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Last modified February 20 22:25 EST 2018. Contains 299387 sequences. (Running on oeis4.)