I'm calculating the product of (n(n-1)-2i), n fixed, i varying from 1 to k, and I would like to express it as a sum, using a kind of Newton's formula. The problem is, the coefficients are a bit more complicated than the binomial ones, there is the (-2) that is always there, but the i changes, so that I would need to be able to calculate each time the sum of all the possible products of i distinct terms in 1,2,...,k.

For example, for k=4, I would have :

i=1 : 1 + 2 + 3 + 4 = 10.

i=2 : 1x2 + 1x3 + 1x4 + 2x3 + 2x4 + 3x4 = 35

i=3 : 1x2x3 + 1x2x4 + 1x3x4 + 2x3x4 = 50

i=4 : 1x2x3x4 = 24

Is there a known formula, or at least a name, for those kind of coefficients  ? Thank you in advance and sorry if it's not clear.

I'm calculating

$$\prod _{i=1} ^k\big( n(n-1)-2i \big)$$

for $n$ fixed, and I would like to express it as a sum, using a kind of Newton's formula. The problem is, the coefficients are a bit more complicated than the binomial ones, there is the $-2$ that is always there, but $i$ changes, so that I would need to be able to calculate each time the sum of all the possible products of $i$ distinct terms in $1,2, \dots, k$.

For example, for $k=4$, I would have :

$$\begin{eqnarray} i=1 &:& \quad 1 + 2 + 3 + 4 &=& 10 \\ i=2 &:& \quad 1 \cdot 2 + 1 \cdot 3 + 1 \cdot 4 + 2 \cdot 3 + 2 \cdot 4 + 3 \cdot 4 &=& 35 \\ i=3 &:& \quad 1 \cdot 2 \cdot 3 + 1 \cdot 2 \cdot 4 + 1 \cdot 3 \cdot 4 + 2 \cdot 3 \cdot 4 &=& 50 \\ i=4 &:& \quad 1 \cdot 2 \cdot 3 \cdot 4 &=& 24 \end{eqnarray}$$

Is there a known formula, or at least a name, for those kind of coefficients? Thank you in advance and sorry if it's not clear.

I'm calculating the product of (n(n-1)-2i), n fixed, i varying from 1 to k, and I would like to express it as a sum, using a kind of Newton's formula. The problem is, the coefficients are a bit more complicated than the binomial ones, there is the (-2) that is always there, but the i changes, so that I would need to be able to calculate each time the sum of all the possible products of i distinct terms in 1,2,...,k.

For example, for k=4, I would have :

i=1 : 1+2+3+4 = 10.

i=2 : 1x2 + 1x3 + 1x4 + 2x3 + 2x4 + 3x4 = 35

i=3 : 1x2x3 + 1x2x4 + 2x3x4 = 38

i=4 : 1x2x3x4 = 24

Is there a known formula, or at least a name, for those kind of coefficients ? Thank you in advance and sorry if it's not clear.

I'm calculating the product of (n(n-1)-2i), n fixed, i varying from 1 to k, and I would like to express it as a sum, using a kind of Newton's formula. The problem is, the coefficients are a bit more complicated than the binomial ones, there is the (-2) that is always there, but the i changes, so that I would need to be able to calculate each time the sum of all the possible products of i distinct terms in 1,2,...,k.

For example, for k=4, I would have :

i=1 : 1 + 2 + 3 + 4 = 10.

i=2 : 1x2 + 1x3 + 1x4 + 2x3 + 2x4 + 3x4 = 35

i=3 : 1x2x3 + 1x2x4 + 1x3x4 + 2x3x4 = 50

i=4 : 1x2x3x4 = 24

Is there a known formula, or at least a name, for those kind of coefficients ? Thank you in advance and sorry if it's not clear.