$$Define:{\nabla ^0}f(n) = f(n),f(N) = \sum\limits_{n = 0}^{N - 1} {{\nabla ^1}f(n + 1)} ,{\nabla ^{ - 1}}f(N) = \sum\limits_{n = 0}^{N - 1} {f(n + 1)} $$
$${\mathop{\rm Re}\nolimits} {\rm{cursive}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} define{\kern 1pt} {\kern 1pt} {\kern 1pt} {\nabla ^p},p \in Z$$
$$Define:SUM(N,[{K_1}],[{T_1} = 1]) = \sum\limits_{n = 0}^{N - 1} {({K_1} + } n)$$
$$SUM(N,[{K_1},{K_2}],[{T_1} = 1,{T_2} = {T_1} + 2 - p]) = \sum\limits_{n = 0}^{N - 1} {({K_2} + } n){\nabla ^p}SUM(n + 1,[{K_1}],[{T_1}])$$
$${\rm{Re}}cursive\,\,\,define\,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} SUM(N,[{K_1},{K_2}...{K_M}],[{T_1},{T_2}...{T_M}])$$
They are nested sums:
$$SUM(N,[{K_1},{K_2},{K_3}],[1,2,3]) = \sum\limits_{n = 0}^{N - 1} {({K_1} + } n)({K_2} + n)({K_3} + n)$$
$$SUM(N,[{K_1},{K_2},{K_3}],[1,3,5]) = \sum\limits_{{n_3} = 0}^{N - 1} {({K_3} + } {n_3})\sum\limits_{{n_2} = 0}^{{n_2}} {({K_2} + } {n_2})\sum\limits_{{n_1} = 0}^{{n_2}} {({K_1} + } {n_1})$$
$$SUM(N,[1,1,1...1],[1,2,3...M]) = {1^M} + {2^M} + ... + {N^M}$$
$$SUM(N,[1,1,1...1],[1,3,5...2M - 1]) = {S_2}(N + M,N) = \sum\limits_{1 \le {i_1} \le {i_2} \le ... \le {i_M} \le N} {{i_1}{i_2}...{i_M}} $$
$$SUM(N,[1,2,3...M],[1,3,5...2M - 1]) = {S_1}(N + M,N) = \sum\limits_{1 \le {i_1} \prec {i_2} \prec ... \prec {i_M} \le N + M - 1} {{i_1}{i_2}...{i_M}} $$
$${\rm{SUM}}(n - k + 1,[1,2,3...k],[1,3,5...2k - 1]){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} can{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} do{\kern 1pt} {\kern 1pt} {\kern 1pt} this{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{calculation}}{\kern 1pt} {\kern 1pt} {\kern 1pt} $$
The calculation result is:
$$SUM(N,[{K_1},{K_2}...{K_M}],[{T_1},{T_2}...{T_M}]) = \sum\limits_{g = 0}^M {H(g)\left( {_{{T_M} - M + 1 + g}^{N + {T_M} - M}} \right)} $$
To obtain H(g), use an auxiliary form to calculate.
$$({K_1} + {T_1})({K_2} + {T_2})...({K_M} + {T_M}) = \sum {\coprod\limits_{i = 1}^M {{X_i}} } ,{X_i} = {K_i}{\kern 1pt} {\kern 1pt} or{\kern 1pt} {\kern 1pt} {\kern 1pt} {T_i}$$
Define:
$$X(T) = count{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {X_i} \in \{ {T_1},{T_2},{T_3}...{T_M}\} $$
$${X_{T - 1}} = count{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of\{ {X_1},{X_2}...{X_{i - 1}}\} \in \{ {T_1},{T_2}...{T_{i - 1}}\} $$
$${X_{K - 1}} = count{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of\{ {X_1},{X_2}...{X_{i - 1}}\} \in \{ {K_1},{K_2}...{K_{i - 1}}\} $$
$${X_{T - 1}} + {X_{K - 1}} = i - 1$$
Don't swap the factors in the ∏Xi,then each ∏Xi corresponds to one expression in the sum.
$$H(g) = \sum\limits_{X(T) = g} {\prod\limits_{i = 1}^M {{B_i}} } .If{\kern 1pt} {\kern 1pt} {X_i} = {T_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} then{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {B_i} = {T_i} - {X_{K - 1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} else{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {B_i} = {K_i} + {X_{T - 1}}$$
For example,SUM(N,[1,2,3],[1,3,5]):
$$Form = ({K_1} + {T_1})({K_2} + {T_2})({K_3} + {T_3}) = (1 + {T_1})(2 + {T_2})(3 + {T_3})$$
$$\sum\limits_{X(T) = 0} {\prod {{X_i}} } = {K_1}{K_2}{K_3} \to H(0) = 1 \times 2 \times 3 = 6$$
$$\sum\limits_{X(T) = 1} {\prod {{X_i}} } = {T_1}{K_2}{K_3} + {K_1}{T_2}{K_3} + {K_1}{K_2}{T_3} \to $$
$$\eqalign{
& H(1) = {T_1}(2 + {X_{T - 1}})(3 + {X_{T - 1}}) + {K_1}({T_2} - {X_{K - 1}})({K_3} + {X_{T - 1}}) + {K_1}{K_2}({T_3} - {X_{T - 1}}) \cr
& = 1 \times (2 + 1) \times (3 + 1) + 1 \times (3 - 1) \times (3 + 1) + 1 \times 2 \times (5 - 2) = 26 \cr} $$
$$SUM(N,[1,2,3],[1,3,5]) = 15\left( {_6^{N + 2}} \right) + 35\left( {_5^{N + 2}} \right) + 26\left( {_4^{N + 2}} \right) + 6\left( {_3^{N + 2}} \right)$$