Normal ordering identity for up and down operators ($(D+U{)}^{\ell }={\sum }_{i+j+2m=\ell }\frac{\ell !}{2^{m}i!j!m!}{U}^i{D}^j$)

If $DU-UD=I$, commuting every $D$ past every $U$ gives $(D+U{)}^{\ell }={\sum }_{i+j+2m=\ell }\ell !U^i{D}^j/(2^{m}i!j!m!)$. The recurrence follows from $D{U}^i=U^{i}D+i{U}^{i-1}$. Applied to the empty diagram in the Young lattice, it counts oscillating tableaux.