Problem: Given two positive integers $t,l \in \mathbb{N}$, find a sequence of integers $\{ \gamma_j \}_{j = 1..(t+l)}$ such that for all values $a = 1, \ldots, t+l$, $$(-1)^l \frac{B_{t+l+1-a}}{t+l+1-a} \left( \binom{t}{a} + (-1)^{a+1} \binom{l}{a} \right) = \frac{t!l!}{(t+l+1)!}\sum_{j = a}^{t+l} \frac{\gamma_j}{j!} S(j,a),$$ where above, $S(j,a)$ denotes the Stirling numbers of the first kind.
Note that there is some redundancy in the expression above, since if $a$ exceeds the value of $\max(t,l)$ the the LHS is $0$. Thus for any particular values of $t,l$ the corresponding sequence $\{ \gamma_j \}$ can be padded by $0$'s.
For example, I have computed by hand the squence $\{ \gamma_j \}$ for various values of $t,l$. I have included a few of these sequences below, and am happy to provide more for other values of $t,l$ if anyone would like.
- $t = 2$, $l = 2$ : $\{ -1 \}$
- $t = 2$, $l = 5$ : $\{ -1, -86,-252,-168,0,0,0 \}$
- $t = 4$, $l = 4$ : $\{ -1, 120, 120, 0,0,0,0,0 \}$
- $t = 5$, $l = 5$ : $\{ -1, 726, -1914,-5280,-2640,0,0,0,0,0 \}$
- $t = 5$, $l = 6$ : $\{ -1, 1450, -3564,-28776,-39600,-15840,0,0,0,0,0 \}$
There are many apparent patterns in the sequences, but I would like find a precise formula/expression for the sequence $\{ \gamma_j \}$ for general values of $t,l$?