  The lace expansion is a powerful perturbative technique to analyze the
critical behavior of random spatial processes such as the self-avoiding walk,
percolation and lattice trees and animals. The non-backtracking lace expansion
(NoBLE) is a modification that allows us to improve its applicability in the
nearest-neighbor setting on the $\mathbb Z^d$-lattice for percolation, lattice
trees and lattice animals.
  The NoBLE gives rise to a recursive formula that we study in this paper at a
general level. We state assumptions that guarantee that the solution of this
recursive formula satisfies the infrared bound. In two related papers, we show
that these conditions are satisfied for percolation in $d\geq 11$, for lattice
trees in $d\geq 16$ and for lattice animals in $d\geq 21$.
