|
| |
|
|
A189225
|
|
Entries in a 4 dimensional version of Pascal's triangle: quadrinomial coefficients of (a+b+c+d)^r.
|
|
1
|
|
|
|
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, 4, 6, 4, 1, 1, 5, 5, 5, 10, 20, 20, 10, 20, 10, 10, 30, 30, 30, 60, 30, 10, 30, 30, 10, 5, 20, 20, 30, 60, 30, 20, 60, 60, 20, 5, 20, 30, 20, 5, 1, 5, 5, 10, 20, 10, 10, 30, 30, 10, 5, 20, 30, 20, 5, 1, 5, 10, 10, 5, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,7
|
|
|
COMMENTS
|
The expansion of (a+b+c+d)^r=Sum[Sum[Sum[Binomial[r,p]*Binomial[p,m]*Binomial[m,k]a^(r-n)b^(n-m)c^(m-k)d^k,{k,0,m}],{m,0,p}],{p,0,r}]. Starting at r=0, the (r)th slice of the 4D simplex is a 3D tetrahedron whose sequence starts at a(0) when r=0 and starts at a(n) where n=Binomial[r+3,4]. It has Binomial[r+3,3] terms whose sum is 4^r. The greatest numbers in each 3D tetrahedron form A022917. Also Coefficients S(r,p,m,k) of a, b, c, d in (a+b+c+d)^n can be defined recursively: S(r+1, p, m, k) = S(r, p-1, m-1, k-1)+S(r, p-1, m-1, k)+S(r, p-1, m, k)+S(r, p, m, k) with S(r, p, m, -1) := 0, ...; and S(0, 0, 0, 0)=1. The coefficient S(r, p, m, k) occurs at a(n) in the sequence where n=Binomial[r+3,4]+Binomial[p+2,3]+Binomial[m+1,2]+Binomial[k,1].
|
|
|
LINKS
|
Table of n, a(n) for n=0..125.
|
|
|
FORMULA
|
S(r,p,m,k)=Binomial[r,p]*Binomial[p,m]*Binomial[m,k] and occurs at a(n) in the above sequence where n=Binomial[r+3,4]+Binomial[p+2,3]+Binomial[m+1,2]+Binomial[k,1]
|
|
|
EXAMPLE
|
The 4th slice of this 4D simplex starts at a(35). It comprises a 3D tetrahedron of 35 terms whose sum is 4096. It is organized as follows:
......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 ,4 ,6 ,4 ,1
|
|
|
MATHEMATICA
|
Flatten[Table[Binomial[r, p] Binomial[p, m] Binomial[m, k], {r, 0, 10}, {p, 0, r}, {m, 0, p}, {k, 0, m}]]
|
|
|
PROG
|
Wolfram|Alpha Table[{Binomial[r, p] Binomial[p, m] Binomial[m, k]}, {r, 0, 10}, {p, 0, r}, {m, 0, p}, {k, 0, m}]
|
|
|
CROSSREFS
|
Cf. A007318, A046816
Sequence in context: A136690 A144703 A097026 * A169988 A213259 A067597
Adjacent sequences: A189222 A189223 A189224 * A189226 A189227 A189228
|
|
|
KEYWORD
|
nonn,tabf,easy
|
|
|
AUTHOR
|
Frank M Jackson, Apr 18 2011
|
|
|
STATUS
|
approved
|
| |
|
|
