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A046816
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Pascal's tetrahedron: entries in 3-dimensional version of Pascal triangle, or trinomial coefficients.
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24
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,6
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COMMENTS
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Greatest numbers in each 2D triangle form A022916 (multinomial coefficient n!/([n/3]![(n+1)/3]![(n+2)/3]!).) 2D triangle sums are powers of 3. - Gerald McGarvey, Aug 15 2004
T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), (1,1,0) and (1,1,1). - Dimitri Boscainos, Aug 16 2015
T(n,j,k) is the number of k-dimensional hyperfaces in an n-dimensional hypercube at an edge distance of j from a given vertex. For example, the number of 2D faces in a 3D cube touching a given vertex is T(3,0,2) = 3, and the number of 3D cube 1D edges at a separation of 1 edge from a given vertex is T(3,1,1) = 6. - Eitan Y. Levine, Jul 22 2023
The sums along vertical lines within each slice (when oriented as in the example) give A027907. See "vertical sums" link. - Eitan Y. Levine, May 17 2023
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REFERENCES
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Marco Costantini: Metodo per elevare qualsiasi trinomio a qualsiasi potenza. Archimede, rivista per gli insegnanti e i cultori di matematiche pure e applicate, anno XXXVIII ottobre-dicembre 1986, pp. 205-209. [Vincenzo Librandi, Jul 19 2009]
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LINKS
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FORMULA
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Coefficients of x, y, z in (x+y+z)^n: Let T'(n; i,j,k) := T(n, j,k) where i = n-(j+k). Then T'(n+1; i,j,k) = T'(n; i-1,j,k)+T'(n; i,j-1,k)+T'(n; i,j,k-1), T'(n; i,j,-1) := 0, T'(n; i,j,k) is invariant under permutations of (i,j,k); T'(0, 0, 0)=1.
T'(n;i,j,k) = n!/(i!*j!*k!) and (x+y+z)^n = Sum_{i+j+k=n; 0 <= i,j,k <= n}T'(n; i,j,k)*x^i*y^j*z^k . Hence Sum_{i+j+k=n; 0 <= i,j,k <= n}T'(n; i,j,k} = 3^n. - Gregory Gerard Wojnar, Oct 08 2020
T(n,j,k) = C(n,j) * C(n-j,k), where C(a,b) are the binomial coefficients, elements of A007318. In particular, T(n,j,0) = C(n,j). - Eitan Y. Levine, Jul 22 2023
(-1)^n * Sum_{i=ceiling(n/k),n} (-1)^i * T(i*k,n-i,i) = k^n, for n,k > 0. - Eitan Y. Levine, Aug 31 2023
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EXAMPLE
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The first few slices of the tetrahedron (or pyramid) are:
1
-----------------
1
1 1
-----------------
1
2 2
1 2 1
-----------------
1 .... Here is the third slice of the pyramid
3 3
3 6 3
1 3 3 1
----------------
...
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MAPLE
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p:= proc(i, j, k) option remember;
if k<0 or i<0 or i>k or j<0 or j>i then 0
elif {i, j, k}={0} then 1
else p(i, j, k-1) +p(i-1, j, k-1) +p(i-1, j-1, k-1)
fi
end:
seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..10);
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MATHEMATICA
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p[i_, j_, k_] := p[i, j, k] = Which[ k<0 || i<0 || i>k || j<0 || j>i, 0, {i, j, k} == {0, 0, 0}, 1, True, p[i, j, k-1] + p[i-1, j, k-1] + p[i-1, j-1, k-1]]; Table[p[i, j, k], {k, 0, 6}, {i, 0, k}, {j, 0, i}] // Flatten (* Jean-François Alcover, Dec 31 2012, translated from Alois P. Heinz's Maple program *)
(* or *)
Flatten[CoefficientList[CoefficientList[CoefficientList[Series[1/(1-x-x*y-x*y*z), {x, 0, 6}], x], y], z]] (* Georg Fischer, May 29 2019 *)
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PROG
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(Haskell)
a046816 n = a046816_list !! n
a046816_list = concat $ concat $ iterate ([[1], [1, 1]] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
(p:ps) + (q:qs) = p + q : ps + qs
ps + qs = ps ++ qs
(p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
_ * _ = []
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CROSSREFS
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Entry [3, 2] in each slice gives A002378, entry [4, 3] in each slice gives A027480, entry [5, 2] in each slice gives A033488, entry [5, 3] in each slice gives A033487.
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KEYWORD
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AUTHOR
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STATUS
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approved
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