There are no approved revisions of this page, so it may
not have been
reviewed.
This article page is a stub, please help by expanding it.
The are two versions of [
nomial] multinomial coefficients: [
variate] or [univariate].
[kvariate knomial] multinomial coefficients
The
[variate nomial] multinomial coefficients
(n1, n2, ..., nk )! := (n1 + n2 + ⋯ + nk 
n1, n2, ..., nk 

) := (n1 + n2 + ⋯ + nk )!  n1! n2! ⋯ nk !  , k ≥ 1, 
where
is a
factorial, are a generalization of the [
2variate
2nomial]
binomial coefficients for the [
variate
nomial]
multinomials
 ${\begin{array}{l}\displaystyle {{\left(\sum _{i=1}^{k}x_{i}\right)^{n}=}\sum _{{n_{i}\geq 0,\,1\leq i\leq k} \atop {\sum _{i=1}^{k}\!n_{i}{\scriptstyle =}n}}(n_{1},\,n_{2},\,\ldots ,\,n_{k})!\;\prod _{i=1}^{k}{x_{i}}^{n_{i}},\quad k\geq 1.}\end{array}}$
For the degenerate case
(“[
1variate] monomial coefficients”!), we have
.
[Univariate knomial] multinomial coefficients
 For [univariate nomial] multinomial coefficients, see: integer compositions into n parts of size at most m.
The
[univariate nomial] multinomial coefficients,
, given by the recurrence relation


Ck (n , j ) := Ck (n − 1, i ), n ≥ 1, 0 ≤ j ≤ (k − 1) n, 
where
for
or
, are a generalization of the [univariate
2nomial]
binomial coefficients for the [univariate
nomial]
multinomials
 .
For the degenerate case
(“[univariate] monomial coefficients”!), we have
.
Trinomial coefficients
Trivariate trinomial coefficients
The [trivariate] trinomial coefficients
appear in the series expansion of the
th power of the trivariate trinomial
${\begin{array}{l}\displaystyle {(x_{1}+x_{2}+x_{3})^{n}=\sum _{{n_{1},\,n_{2},\,n_{3}\geq 0} \atop {n_{1}+n_{2}+n_{3}=n}}{\binom {n}{n_{1},n_{2},n_{3}}}\,{x_{1}}^{n_{1}}\,{x_{2}}^{n_{2}}\,{x_{3}}^{n_{3}}.}\end{array}}$
The [trivariate] trinomial coefficients, where
is the layer,
is the row, and
is the column, are obtained by the recurrence relation


(n , j , k )! := (n − 1, j − 1, k − 1)! + (n − 1, j − 1, k )! + (n − 1, j , k )!, n ≥ 1, 0 ≤ j ≤ n, 0 ≤ k ≤ j, 
where
for
or
or
or
.
The [trivariate] trinomial coefficients form a 3dimensional tetrahedral array of coefficients, where each of the
terms of the
th layer is the sum of the 3 closest terms of the
th layer.
Pascal’s tetrahedron
(Pascal’s [triangular] pyramid)^{[1]}
Layer 0 (top layer^{[2]})


Layer 1


Layer 2


Layer 3


Layer 4





1 







4 

4 





6 

12 

6 



4 

12 

12 

4 

1 

4 

6 

4 

1


Layer 5






1 









5 

5 







10 

20 

10 





10 

30 

30 

10 



5 

20 

30 

20 

5 

1 

5 

10 

10 

5 

1


A046816 Pascal’s tetrahedron: entries in 3dimensional version of Pascal’s triangle, or [trivariate] trinomial coefficients.

{1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 3, 6, 3, 1, 3, 3, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 5, 5, 10, 20, 10, 10, 30, 30, 10, 5, 20, 30, 20, 5, 1, 5, 10, 10, 5, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1, ...}
Univariate trinomial coefficients
The [univariate] trinomial coefficients
appear in the series expansion of the
th power of the [univariate] trinomial

(x 1 + x 2 + x 3 ) n = x n (x 0 + x 1 + x 2 ) n = x n C3(n, j ) x j. 
The [univariate] trinomial coefficients are obtained by the recurrence relation


C3(n , j ) := C3(n − 1, i ), n ≥ 1, 0 ≤ j ≤ 2 n, 
where
for
or
.
For [univariate] trinomial coefficient array, row
is the sequence of coefficients of
in
.
Row is the sequence of [univariate] trinomial coefficients of in


Anumber

0

1

A??????

1

1, 1, 1

A??????

2

1, 2, 3, 2, 1

A??????

3

1, 3, 6, 7, 6, 3, 1

A??????

4

1, 4, 10, 16, 19, 16, 10, 4, 1

A??????

5

1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1

A??????

6

1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1

A??????

7

1, 7, 28, 77, 161, 266, 357, 393, 357, 266, 161, 77, 28, 7, 1

A??????

A027907 Triangle of [univariate] trinomial coefficients. Row
is the sequence of coefficients of
.

{1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 10, 16, 19, 16, 10, 4, 1, 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1, 1, 7, 28, 77, 161, 266, 357, 393, 357, 266, 161, 77, 28, 7, 1, 1, 8, 36, 112, 266, ...}
Quadrinomial coefficients
Quadrivariate quadrinomial coefficients
A189225 Entries in a 4dimensional version [4dimensional simplex: pentachoron] of Pascal’s triangle: [quadrivariate] quadrinomial coefficients of
.

{1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 3, 3, 3, 3, 6, 6, 3, 6, 3, 1, 3, 3, 3, 6, 3, 1, 3, 3, 1, 1, 4, 4, 4, 6, 12, 12, 6, 12, 6, 4, 12, 12, 12, 24, 12, 4, 12, 12, 4, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, ...}
Univariate quadrinomial coefficients
See: Tetrahedral dice.
A008287 Triangle of [univariate] quadrinomial (also called tetranomial) coefficients. Row
is the sequence of coefficients of
.

{1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1, 1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1, ...}
Notes
External links