How to check these terms in $\int{x^x dx}$ power series are correct?

I think the situation can be simplified. Start with $$ f := \sum_{n=1}^\infty x^n \sum_{k=0}^{n-1} B(n,k)\, y^k. \tag{1}$$ Then use the total derivative to get $$ D(n,k) := D[ x^n y^k ] = x^{n-1}y^{k-1}(k+n\,y) \tag{2} $$ after using $\, D[ y ]=1/x.\,$ The derivative splits into $$ D_1(n,k) := kx^{n-1}y^{k-1},\, D_2(n,k) := nx^{n-1}y^k. \tag{3} $$ Now a bit of algebra shows that $$ -B(n,k)D_1(n,k)=B(n,k-1)D_2(n,k-1). \tag{4} $$ The minus sign leads to cancellation in the derivative of the sum in equation $(1)$ and simplifies into $\,D[ f ] = \exp(x\,y).\,$ Replacing $\,y\,$ with $\,\log(x)\,$ gives $\,x^x.\,$