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, 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

Offset: 0

Author

Frank M Jackson, Apr 18 2011

Keywords

• nonn
• tabf
• easy

Comments

The expansion of (a + b + c + d)^r = Sum_{p=0..r} Sum_{m=0..p} Sum_{k=0..m} binomial(r,p)*binomial(p,m)*binomial(m,k)*a^(r-n)*b^(n-m)*c^(m-k)*d^k. 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, the 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).

T(n,i,j,k) is the number of lattice paths from (0,0,0,0) to (n,i,j,k) with steps (1,0,0,0), (1,1,0,0), (1,1,1,0) and (1,1,1,1). - Dimitri Boscainos, Aug 16 2015

Numbers of ways to classify n circles black, red, green, or yellow; classified first by how many circles there are altogether, then by how many are of each color. - J. Lowell, Nov 17 2024

Examples

			The 4th slice of this 4D simplex starts at a(35). It comprises a 3D tetrahedron of 35 terms whose sum is 256. 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
		

Crossrefs

Cf. A007318 (Pascal's triangle), A046816 (Pascal's tetrahedron).

Programs

Mathematica

Flatten[Table[Binomial[r,p] Binomial[p,m] Binomial[m,k], {r,0,10}, {p,0,r}, {m,0,p}, {k,0,m}]] (* or *)
Flatten[CoefficientList[CoefficientList[CoefficientList[CoefficientList[Series[1/(1-w-w*x-w*x*y-w*x*y*z), {w, 0, 5}], w], x], y], z]] (* Georg Fischer, Jun 01 2019 *)

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).

G.f.: 1/(1-w-w*x-w*x*y-w*x*y*z). - Georg Fischer, Jun 01 2019

Extensions

Example corrected by Dimitri Boscainos, Aug 16 2015