You can also develop the selfpower \( x^x \) into a powerseries at 1:
\(
x^x = \sum_{n=0}^\infty a_n (x-1)^n,\quad |x|<1\\
a_n = \frac{1}{n!}\sum_{k=0}^n \operatorname{lc}(n,k)
\)
where lc is the so called Lehmer-Comtet numbers:
\( \operatorname{lc}(n,k) = \sum_{l=k}^n \left(l\\ k\right) k^{l-k} s(n,l) \)
where \( s(n,l) \) are the stirling numbers of the first kind.
The sequence of the first few derivatives ( \( a_n n! \)) is
1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, ... (see sloane A005727)
The coefficients of the indefinite integral at 1 are then:
\( \frac{1}{n!}\sum_{k=0}^{n-1} \operatorname{lc}(n-1,k) \)
accordingly the sequence of derivatives of the integral is shifted 1 to the right.
\(
x^x = \sum_{n=0}^\infty a_n (x-1)^n,\quad |x|<1\\
a_n = \frac{1}{n!}\sum_{k=0}^n \operatorname{lc}(n,k)
\)
where lc is the so called Lehmer-Comtet numbers:
\( \operatorname{lc}(n,k) = \sum_{l=k}^n \left(l\\ k\right) k^{l-k} s(n,l) \)
where \( s(n,l) \) are the stirling numbers of the first kind.
The sequence of the first few derivatives ( \( a_n n! \)) is
1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, ... (see sloane A005727)
The coefficients of the indefinite integral at 1 are then:
\( \frac{1}{n!}\sum_{k=0}^{n-1} \operatorname{lc}(n-1,k) \)
accordingly the sequence of derivatives of the integral is shifted 1 to the right.