A258773 Triangle read by rows, T(n,k) = (-1)^(n-k)*C(n,k)*k^n, for n >= 0 and 0 <= k <= n.

1, 0, 1, 0, -2, 4, 0, 3, -24, 27, 0, -4, 96, -324, 256, 0, 5, -320, 2430, -5120, 3125, 0, -6, 960, -14580, 61440, -93750, 46656, 0, 7, -2688, 76545, -573440, 1640625, -1959552, 823543, 0, -8, 7168, -367416, 4587520, -21875000, 47029248, -46118408, 16777216

Offset: 0

Author

Peter Luschny, Jun 09 2015

Keywords

• sign
• easy
• tabl

Comments

The row polynomials are p(0, x) = 1, and p(n, x) = Eu(x)^n (x-1)^n, for n >= 1, where Eu(x) := x*d/dx is the Euler derivative with respect to x. See A075513. - Wolfdieter Lang, Oct 12 2022

Coefficients of the Sidi polynomials (-1)^n*x*D_{n,0,n}(x), for n >= 0, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980]. - Wolfdieter Lang, Apr 10 2023

Examples

			Triangle begins:
  [1]
  [0,  1]
  [0, -2,     4]
  [0,  3,   -24,     27]
  [0, -4,    96,   -324,     256]
  [0,  5,  -320,   2430,   -5120,    3125]
  [0, -6,   960, -14580,   61440,  -93750,    46656]
  [0,  7, -2688,  76545, -573440, 1640625, -1959552, 823543]
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1326
• Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Crossrefs

Cf. A000142, A000312, A072034, A075513, A154715.

Programs

Maple

seq(seq((-1)^(n-k)*binomial(n, k)*k^n, k=0..n), n=0..8);
T_row := proc(n) (-1)^n*(1-exp(x))^n/n!; diff(%,[x$n]); subs(exp(x)=t, n!*expand(%,x)); CoefficientList(%,t) end: seq(print(T_row(n)), n=0..7);

Mathematica

Flatten@Table[(-1)^(n - k) Binomial[n, k] k^n, {n, 0 , 10}, {k, 0, n}] (* G. C. Greubel, Dec 17 2015 *)

Formula

Sum_{k=0..n} T(n,k) = n!.

Sum_{k=0..n} |T(n,k)| = A072034(n).

Sum_{n>=0} Sum_{k=0..n} T(n,k) x^k y^n/n! = 1/(1 + W(-x*y*exp(-y))) where W is the Lambert W function. - Robert Israel, Dec 16 2015

T(n,n) = A000312(n). - Peter Luschny, Dec 17 2015

T(n, k+1) = n * A075513(n, k) if n > 0. - Michael Somos, May 13 2018