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Multinomial coefficients

The are two versions of [
 k
-nomial] multinomial coefficients: [
 k
-variate] or [univariate].

[k-variate k-nomial] multinomial coefficients

The [
 k
-variate
 k
-nomial] multinomial coefficients

(n1, n2, ..., nk  )!  :=(
 n1 + n2 + ⋯ + nk n1, n2, ..., nk
):=
 (n1 + n2 + ⋯ + nk  )! n1! n2! ⋯ nk !
, k  ≥ 1,

where
 n!
is a factorial, are a generalization of the [2-variate 2-nomial] binomial coefficients for the [
 k
-variate
 k
-nomial] multinomials
${\displaystyle {\begin{array}{l}\displaystyle {{\left(\sum _{i=1}^{k}x_{i}\right)^{n}=}\sum _{{n_{i}\geq 0,\,1\leq i\leq k} \atop {\sum _{i=1}^{k}\!n_{i}{\scriptstyle =}n}}(n_{1},\,n_{2},\,\ldots ,\,n_{k})!\;\prod _{i=1}^{k}{x_{i}}^{n_{i}},\quad k\geq 1.}\end{array}}}$
For the degenerate case
 k  = 1
(“[1-variate] monomial coefficients”!), we have
 (n1 )! = 1
.

[Univariate k-nomial] multinomial coefficients

For [univariate
 m
-nomial] multinomial coefficients, see: integer compositions into n parts of size at most m.
The [univariate
 k
-nomial] multinomial coefficients
,
 k   ≥   2
, given by the recurrence relation
 Ck (0, 0)  :=  1;
Ck (n ,   j )  :=
 j ∑ i  =  j − (k  − 1) n

Ck (n − 1, i ), n ≥ 1, 0 ≤   j ≤ (k  − 1) n,
where
 Ck (n,   j )   :=   0
for
 j < 0
or
 j > (k  −  1) n
, are a generalization of the [univariate 2-nomial] binomial coefficients for the [univariate
 k
-nomial] multinomials
 k   − 1 ∑ i  = 0

xi
n  =
 (k   − 1) n ∑ j  = 0

Ck (n ,   j ) x   j
.
For the degenerate case
 k  = 1
(“[univariate] monomial coefficients”!), we have
 Ck (n, 0) = 1
.

Trinomial coefficients

Trivariate trinomial coefficients

The [trivariate] trinomial coefficients
 (n1, n2, n3 )!
appear in the series expansion of the
 n
th power of the trivariate trinomial

${\displaystyle {\begin{array}{l}\displaystyle {(x_{1}+x_{2}+x_{3})^{n}=\sum _{{n_{1},\,n_{2},\,n_{3}\geq 0} \atop {n_{1}+n_{2}+n_{3}=n}}{\binom {n}{n_{1},n_{2},n_{3}}}\,{x_{1}}^{n_{1}}\,{x_{2}}^{n_{2}}\,{x_{3}}^{n_{3}}.}\end{array}}}$

The [trivariate] trinomial coefficients, where
 n
is the layer,
 j
is the row, and
 k
is the column, are obtained by the recurrence relation
 (0, 0, 0)!  :=  1;
 (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
 (n,   j , k )!   :=   0
for
 j < 0
or
 j > n
or
 k < 0
or
 k >   j
.
The [trivariate] trinomial coefficients form a 3-dimensional tetrahedral array of coefficients, where each of the
 tn  + 1
terms of the
 n
th layer is the sum of the 3 closest terms of the
 (n  −  1)
th layer.
Pascal’s tetrahedron
(Pascal’s [triangular] pyramid)[1]
Layer 0
(top layer[2])
 1
Layer 1
 1 1 1
Layer 2
 1 2 2 1 2 1
Layer 3
 1 3 3 3 6 3 1 3 3 1
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 3-dimensional 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
 C3(n,   j )
appear in the series expansion of the
 n
th power of the [univariate] trinomial
(x 1 + x 2 + x 3 )n  =  xn (x 0 + x 1 + x 2 )n  =  xn
 2n ∑ j  = 0

C3(n,   j ) x  j.

The [univariate] trinomial coefficients are obtained by the recurrence relation

 C3(0, 0)  :=  1;
C3(n ,   j )  :=
 j ∑ i  = j − 2 n

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

 i  = 0
xi  )n
.

Row
 n, n   ≥   0,
is the sequence of [univariate] trinomial coefficients of
 x  j, 0   ≤   j   ≤   2 n ,
in
 (x 0 + x 1 + x 2 ) n

 n
A-number
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
 n, n   ≥   0,
is the sequence of coefficients of
 (1 + x + x 2 ) n
.
{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, ...}

A189225 Entries in a 4-dimensional version [4-dimensional simplex: pentachoron] of Pascal’s triangle: [quadrivariate] quadrinomial coefficients of
 (x1 + x2 + x3 + x4 ) n
.
{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, ...}