cp's OEIS Frontend

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

%I A189225 #44 Nov 22 2024 11:09:14
%S A189225 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,
%T A189225 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,
%U A189225 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
%N A189225 Entries in a 4-dimensional version of Pascal's triangle: quadrinomial coefficients of (a + b + c + d)^r.
%C A189225 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).
%C A189225 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
%C A189225 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
%F A189225 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).
%F A189225 G.f.: 1/(1-w-w*x-w*x*y-w*x*y*z). - _Georg Fischer_, Jun 01 2019
%e A189225 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:
%e A189225           1
%e A189225 .
%e A189225           4
%e A189225         4   4
%e A189225 .
%e A189225           6
%e A189225        12  12
%e A189225       6  12   6
%e A189225 .
%e A189225           4
%e A189225        12  12
%e A189225      12  24  12
%e A189225     4  12  12   4
%e A189225 .
%e A189225           1
%e A189225         4   4
%e A189225       6  12   6
%e A189225     4  12  12   4
%e A189225   1   4   6   4   1
%t A189225 Flatten[Table[Binomial[r,p] Binomial[p,m] Binomial[m,k], {r,0,10}, {p,0,r}, {m,0,p}, {k,0,m}]] (* or *)
%t A189225 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 *)
%Y A189225 Cf. A007318 (Pascal's triangle), A046816 (Pascal's tetrahedron).
%K A189225 nonn,tabf,easy
%O A189225 0,7
%A A189225 _Frank M Jackson_, Apr 18 2011
%E A189225 Example corrected by _Dimitri Boscainos_, Aug 16 2015