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A189225 Entries in a 4-dimensional version of Pascal's triangle: quadrinomial coefficients of (a + b + c + d)^r. 5
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_{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

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 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

MATHEMATICA

Flatten[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: A276825 A270644 A097026 * A169988 A213259 A067597

Adjacent sequences:  A189222 A189223 A189224 * A189226 A189227 A189228

KEYWORD

nonn,tabf,easy

AUTHOR

Frank M Jackson, Apr 18 2011

EXTENSIONS

Example corrected by Dimitri Boscainos, Aug 16 2015

STATUS

approved

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Last modified December 10 21:23 EST 2016. Contains 279011 sequences.