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The are two versions of [
-nomial] multinomial coefficients: [
-variate] or [univariate].
[k-variate k-nomial] 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 [
2-variate
2-nomial]
binomial coefficients for the [
-variate
-nomial]
multinomials
-
For the degenerate case
(“[
1-variate] monomial coefficients”!), we have
.
[Univariate k-nomial] 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
2-nomial]
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
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 3-dimensional 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])
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Layer 1
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Layer 2
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Layer 3
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Layer 4
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12 |
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1
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Layer 5
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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
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
|
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A-number
|
0
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1
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A??????
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1
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1, 1, 1
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A??????
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2
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1, 2, 3, 2, 1
|
A??????
|
3
|
1, 3, 6, 7, 6, 3, 1
|
A??????
|
4
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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 4-dimensional version [4-dimensional 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