# Questions tagged [multinomial-coefficients]

For questions related to multinomial coefficients, a generalization of binomial coefficients.

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### Number of k-tuples of non-negative integers whose sum equals a given integer

Does the sum over the non-negative integers, $$\sum\limits_{ {i_1, \ldots i_k \geq 0:\\\ i_1+\ldots i_k=L }} 1$$ have a closed expression, where $L$ and $k$ are some integers?
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### Multinomial sum with positive coefficients

Consider the following sum. $$\sum_{ \substack{L_1 L_2 ... L_k :\\ L_1 +... + L_k = N\\ \forall i \, \, \, L_i > 0 }}\binom{N}{L_1 , L_2 , ... L_k}$$ It is well known from the Multinomial ...
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### A way to find a closed form of multinomial convoluted polynome.

It has been a couple of days since I stepped into some hard formula of convoluted polynomial of the form: $$\sum_k \binom{n}{k}\binom{m}{m-k} x^k$$ I tried to dabble with convolution proofs of ...
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### Is this type of factorisation always possible?

$x^2 + xy + xz + yz$ can be factorised as $(x+y)(x+z)$. Is there a simple formula for factorising $ax^2 + bxy + cxz + dyz$ into $(a'x+b'y)(a'x+c'z)$, how can I get $a',b',c'$? In general : Given a ...