A185164 Coefficients of a set of polynomials associated with the derivatives of x^x.

1, 1, 2, 3, 6, 10, 24, 40, 15, 120, 196, 105, 720, 1148, 700, 105, 5040, 7848, 5068, 1260, 40320, 61416, 40740, 12600, 945, 362880, 541728, 363660, 126280, 17325, 3628800, 5319072, 3584856, 1332100, 242550, 10395, 39916800, 57545280, 38764440, 15020720, 3213210, 270270

Offset: 2

Author

Peter Bala, Mar 12 2012

Keywords

• nonn
• easy
• tabf

Comments

Gould shows that the derivatives of x^x are given by (d/dx)^n(x^x) = (x^x)*Sum_{k = 0..n} (-1)^k*binomial(n,k)*(1 + log(x))^(n-k)*x^(-k)*R(k,x), where R(n,x) is a polynomial in x of degree floor(n/2). The first few values are R(0,x) = 1, R(1,x) = 0, R(2,x) = x, R(3,x) = x and R(4,x) = 2*x + 3*x^2. The coefficients of these polynomials are listed in the table for n >= 2. Gould gives an explicit formula for R(n,x) as a triple sum, and also an expression in terms of the Comtet numbers A008296.

This table read by diagonals gives A075856.

Examples

			Triangle begins
n\k.|.....1.....2.....3.....4
= = = = = = = = = = = = = = =
..2.|.....1
..3.|.....1
..4.|.....2.....3
..5.|.....6....10
..6.|....24....40....15
..7.|...120...196...105
..8.|...720..1148...700...105
..9.|..5040..7848..5068..1260
...
Fourth derivative of x^x:
x^(-x)*(d/dx)^4(x^x) = (1+log(x))^4 + C(4,2)/x^2*(1+log(x))^2*x - C(4,3)/x^3*(1+log(x)) + C(4,4)/x^4*(2*x + 3*x^2).
Example of recurrence relation for table entries:
T(7,2) = 4*T(6,2) + 6*T(5,1) = 4*40 + 6*6 = 196.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10001 (rows 2 to 200, flattened)
• Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
• H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.

Crossrefs

Cf. A008296, A075856, A203852 (row sums).

Programs

Maple

T[2,1]:= 1:
for n from 3 to 15 do
  for k from 1 to floor(n/2) do
    T[n,k]:= (n-1-k)*`if`(k<= floor((n-1)/2),T[n-1,k],0) + `if`(k>=2 and k-1 <= floor((n-2)/2),(n-1)*T[n-2,k-1],0)
od od:
seq(seq(T[n,k],k=1..floor(n/2)),n=2..15); # Robert Israel, Jan 13 2016

Mathematica

m = 14; F = Exp[t (x + (1-x) Log[1-x])];
cc = CoefficientList[# + O[t]^m, t]& /@ CoefficientList[F + O[x]^m, x]* Range[0, m - 1]!;
Rest /@ Drop[cc, 2] (* Jean-François Alcover, Jun 26 2019 *)

Sage

# uses[bell_transform from A264428]
# Computes the full triangle for n>=0 and 0<=k<=n.
def A185164_row(n):
    g = lambda k: factorial(k-1) if k>0 else 0
    s = [g(k) for k in (0..n)]
    return bell_transform(n, s)
[A185164_row(n) for n in (0..10)] # Peter Luschny, Jan 13 2016

Formula

Recurrence relation: T(n+1,k) = (n - k)*T(n,k) + n*T(n-1,k-1).

The diagonal entries D(n,k) := T(n+k,k) satisfy the recurrence D(n+1,k) = n*D(n,k) + (n + k)*D(n,k-1) so this table read by diagonals is A075856.

E.g.f.: F(x,t) = exp(t*(x + (1 - x)*log(1 - x))) = Sum_{n = 0..oo} R(n,t)*x^n/n! = 1 + t*x^2/2! + t*x^3/3! + (2*t + 3*t^2)*x^4/4! + .... The e.g.f. F(x,t) satisfies the partial differential equation (1 - x)*dF/dx + t*dF/dt = x*t*F.

This gives the recurrence relation for the row generating polynomials: R(n+1,x) = n*R(n,x) - x*d/dx(R(n,x)) + n*x*R(n-1,x) for n >= 1, with initial conditions R(0,x) = 1, R(1,x) = 0.

The e.g.f. for the triangle read by diagonals is given by the series reversion (with respect to x) (x - t*(x + (1 - x)*log(1 - x)))^(-1) = x + t*x^2/2! + (t + 3*t^2)x^3/3! + (2*t + 10*t^2 + 15*t^3)*x^4/4! + ....

Diagonal sums: Sum_{k = 1..n} T(n+k,k) = n^n , n >= 1.

Row sums A203852.

Also the Bell transform of the sequence g(k) = (k-1)! if k>0 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 13 2016

Extensions

More terms from Jean-François Alcover, Jun 26 2019