${m+j \choose M} {1-M \choose m+i-M}$ 的 $M$ 的部分总和?

$$ \sum_{m' \geq m} {m+j \choose m'} {B - m' \choose m+i-m'} $$ 是否有(很可能有)任何类型的公式?

我主要对 $B=1$ (或素数)的情况感到好奇,如果那是任何不那么复杂的类型。

编辑:按照建议,下面是 $m=0\ldots 4$、$i,j< 10$ 的 $B=1$ 响应,组织为矩阵(每个 $m$ 一个)。

$m=0: \begin{pmatrix}{2}& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ {2}& {2}& 1& 1& 1& 1& 1& 1& 1& 1\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {-1}& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& {-2}& 1& 0& 0& 0& 0& 0\\ 0& 0& {-1}& {3}& {-3}& 1& 0& 0& 0& 0\\ 0& 0& 1& {-4}& {6}& {-4}& 1& 0& 0& 0\\ 0& 0& {-1}& {5}& {-10}& {10}& {-5}& 1& 0& 0\\ 0& 0& 1& {-6}& {15}& {-20}& {15}& {-6}& 1& 0\\ 0& 0& {-1}& {7}& {-21}& {35}& {-35}& {21}& {-7}& 1\\ \end{pmatrix}$

$m=1: \begin{pmatrix}{2}& {2}& {3}& {4}& {5}& {6}& {7}& {8}& {9}& {10}\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& {-1}& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& {-2}& 1& 0& 0& 0& 0& 0& 0\\ 0& {-1}& {3}& {-3}& 1& 0& 0& 0& 0& 0\\ 0& 1& {-4}& {6}& {-4}& 1& 0& 0& 0& 0\\ 0& {-1}& {5}& {-10}& {10}& {-5}& 1& 0& 0& 0\\ 0& 1& {-6}& {15}& {-20}& {15}& {-6}& 1& 0& 0\\ 0& {-1}& {7}& {-21}& {35}& {-35}& {21}& {-7}& 1& 0\\ 0& 1& {-8}& {28}& {-56}& {70}& {-56}& {28}& {-8}& 1\\ \end{pmatrix}$

$m=2: \begin{pmatrix}{2}& {3}& {6}& {10}& {15}& {21}& {28}& {36}& {45}& {55}\\ {-2}& {-2}& {-6}& {-10}& {-15}& {-21}& {-28}& {-36}& {-45}& {-55}\\ {2}& 1& {7}& {10}& {15}& {21}& {28}& {36}& {45}& {55}\\ {-2}& 0& {-9}& {-9}& {-15}& {-21}& {-28}& {-36}& {-45}& {-55}\\ {2}& {-1}& {12}& {6}& {16}& {21}& {28}& {36}& {45}& {55}\\ {-2}& {2}& {-16}& 0& {-20}& {-20}& {-28}& {-36}& {-45}& {-55}\\ {2}& {-3}& {21}& {-10}& {30}& {15}& {29}& {36}& {45}& {55}\\ {-2}& {4}& {-27}& {25}& {-50}& 0& {-35}& {-35}& {-45}& {-55}\\ {2}& {-5}& {34}& {-46}& {85}& {-35}& {56}& {28}& {46}& {55}\\ {-2}& {6}& {-42}& {74}& {-141}& {105}& {-112}& 0& {-54}& {-54}\\ \end{pmatrix}$

$m=3: \begin{pmatrix}{2}& {4}& {10}& {20}& {35}& {56}& {84}& {120}& {165}& {220}\\ {-4}& {-7}& {-20}& {-40}& {-70}& {-112}& {-168}& {-240}& {-330}& {-440}\\ {6}& {9}& {31}& {60}& {105}& {168}& {252}& {360}& {495}& {660}\\ {-8}& {-10}& {-44}& {-79}& {-140}& {-224}& {-336}& {-480}& {-660}& {-880}\\ {10}& {10}& {60}& {95}& {176}& {280}& {420}& {600}& {825}& {1100}\\ {-12}& {-9}& {-80}& {-105}& {-216}& {-335}& {-504}& {-720}& {-990}& {-1320}\\ {14}& {7}& {105}& {105}& {266}& {385}& {589}& {840}& {1155}& {1540}\\ {-16}& {-4}& {-136}& {-90}& {-336}& {-420}& {-680}& {-959}& {-1320}& {-1760}\\ {18}& 0& {174}& {54}& {441}& {420}& {792}& {1071}& {1486}& {1980}\\ {-20}& {5}& {-220}& {10}& {-602}& {-350}& {-960}& {-1155}& {-1660}& {-2199}\\ \end{pmatrix}$

$m=4: \begin{pmatrix}{2}& {5}& {15}& {35}& {70}& {126}& {210}& {330}& {495}& {715}\\ {-6}& {-14}& {-45}& {-105}& {-210}& {-378}& {-630}& {-990}& {-1485}& {-2145}\\ {12}& {26}& {91}& {210}& {420}& {756}& {1260}& {1980}& {2970}& {4290}\\ {-20}& {-40}& {-155}& {-349}& {-700}& {-1260}& {-2100}& {-3300}& {-4950}& {-7150}\\ {30}& {55}& {240}& {519}& {1051}& {1890}& {3150}& {4950}& {7425}& {10725}\\ {-42}& {-70}& {-350}& {-714}& {-1477}& {-2645}& {-4410}& {-6930}& {-10395}& {-15015}\\ {56}& {84}& {490}& {924}& {1988}& {3520}& {5881}& {9240}& {13860}& {20020}\\ {-72}& {-96}& {-666}& {-1134}& {-2604}& {-4500}& {-7569}& {-11879}& {-17820}& {-25740}\\ {90}& {105}& {885}& {1323}& {3360}& {5550}& {9495}& {14840}& {22276}& {32175}\\ {-110}& {-110}& {-1155}& {-1463}& {-4312}& {-6600}& {-11715}& {-18095}& {-27236}& {-39324}\\ \end{pmatrix}$

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2022-06-07 14:38:22
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