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

%I A185164 #43 Dec 21 2022 20:14:22
%S A185164 1,1,2,3,6,10,24,40,15,120,196,105,720,1148,700,105,5040,7848,5068,
%T A185164 1260,40320,61416,40740,12600,945,362880,541728,363660,126280,17325,
%U A185164 3628800,5319072,3584856,1332100,242550,10395,39916800,57545280,38764440,15020720,3213210,270270
%N A185164 Coefficients of a set of polynomials associated with the derivatives of x^x.
%C A185164 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.
%C A185164 This table read by diagonals gives A075856.
%H A185164 Robert Israel, <a href="/A185164/b185164.txt">Table of n, a(n) for n = 2..10001</a> (rows 2 to 200, flattened)
%H A185164 Peter Bala, <a href="/A112007/a112007.txt_Bala">Diagonals of triangles with generating function exp(t*F(x)).</a>
%H A185164 H. W. Gould, <a href="http://dx.doi.org/10.1216/rmjm/1181072076">A Set of Polynomials Associated with the Higher Derivatives of y = x^x</a>, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
%F A185164 Recurrence relation: T(n+1,k) = (n - k)*T(n,k) + n*T(n-1,k-1).
%F A185164 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.
%F A185164 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.
%F A185164 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.
%F A185164 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! + ....
%F A185164 Diagonal sums: Sum_{k = 1..n} T(n+k,k) = n^n , n >= 1.
%F A185164 Row sums A203852.
%F A185164 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
%e A185164 Triangle begins
%e A185164 n\k.|.....1.....2.....3.....4
%e A185164 = = = = = = = = = = = = = = =
%e A185164 ..2.|.....1
%e A185164 ..3.|.....1
%e A185164 ..4.|.....2.....3
%e A185164 ..5.|.....6....10
%e A185164 ..6.|....24....40....15
%e A185164 ..7.|...120...196...105
%e A185164 ..8.|...720..1148...700...105
%e A185164 ..9.|..5040..7848..5068..1260
%e A185164 ...
%e A185164 Fourth derivative of x^x:
%e A185164 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).
%e A185164 Example of recurrence relation for table entries:
%e A185164 T(7,2) = 4*T(6,2) + 6*T(5,1) = 4*40 + 6*6 = 196.
%p A185164 T[2,1]:= 1:
%p A185164 for n from 3 to 15 do
%p A185164   for k from 1 to floor(n/2) do
%p A185164     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)
%p A185164 od od:
%p A185164 seq(seq(T[n,k],k=1..floor(n/2)),n=2..15); # _Robert Israel_, Jan 13 2016
%t A185164 m = 14; F = Exp[t (x + (1-x) Log[1-x])];
%t A185164 cc = CoefficientList[# + O[t]^m, t]& /@ CoefficientList[F + O[x]^m, x]* Range[0, m - 1]!;
%t A185164 Rest /@ Drop[cc, 2] (* _Jean-François Alcover_, Jun 26 2019 *)
%o A185164 (Sage) # uses[bell_transform from A264428]
%o A185164 # Computes the full triangle for n>=0 and 0<=k<=n.
%o A185164 def A185164_row(n):
%o A185164     g = lambda k: factorial(k-1) if k>0 else 0
%o A185164     s = [g(k) for k in (0..n)]
%o A185164     return bell_transform(n, s)
%o A185164 [A185164_row(n) for n in (0..10)] # _Peter Luschny_, Jan 13 2016
%Y A185164 Cf. A008296, A075856, A203852 (row sums).
%K A185164 nonn,easy,tabf
%O A185164 2,3
%A A185164 _Peter Bala_, Mar 12 2012
%E A185164 More terms from _Jean-François Alcover_, Jun 26 2019