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%I #111 Sep 29 2023 18:57:24
%S 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,
%T 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,
%U 30,15,20,60,60,20,15,60,90,60,15,6,30,60,60,30,6,1,6,15,20,15,6,1
%N Pascal's tetrahedron: entries in 3-dimensional version of Pascal triangle, or trinomial coefficients.
%C 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
%C 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
%C 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
%C 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
%D 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]
%H Alois P. Heinz, <a href="/A046816/b046816.txt">Table of n, a(n) for n = 0..10659</a>
%H Eitan Y. Levine, <a href="/A046816/a046816_1.txt">Vertical sums</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pascal%27s_pyramid">Pascal's pyramid</a>
%H <a href="/index/3#3DArrays">Index entries for sequences related to 3D arrays of numbers</a>
%F 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.
%F 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
%F G.f.: 1/(1-x-x*y-x*y*z). - _Georg Fischer_, May 29 2019
%F 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
%F (-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
%e The first few slices of the tetrahedron (or pyramid) are:
%e 1
%e -----------------
%e 1
%e 1 1
%e -----------------
%e 1
%e 2 2
%e 1 2 1
%e -----------------
%e 1 .... Here is the third slice of the pyramid
%e 3 3
%e 3 6 3
%e 1 3 3 1
%e ----------------
%e ...
%p p:= proc(i, j, k) option remember;
%p if k<0 or i<0 or i>k or j<0 or j>i then 0
%p elif {i, j, k}={0} then 1
%p else p(i, j, k-1) +p(i-1, j, k-1) +p(i-1, j-1, k-1)
%p fi
%p end:
%p seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..10);
%p # _Alois P. Heinz_, Apr 03 2011
%t 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 *)
%t (* or *)
%t Flatten[CoefficientList[CoefficientList[CoefficientList[Series[1/(1-x-x*y-x*y*z), {x, 0, 6}], x], y],z]] (* _Georg Fischer_, May 29 2019 *)
%o (Haskell)
%o a046816 n = a046816_list !! n
%o a046816_list = concat $ concat $ iterate ([[1],[1,1]] *) [1]
%o instance Num a => Num [a] where
%o fromInteger k = [fromInteger k]
%o (p:ps) + (q:qs) = p + q : ps + qs
%o ps + qs = ps ++ qs
%o (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
%o _ * _ = []
%o -- _Reinhard Zumkeller_, Apr 02 2011
%Y Cf. A007318, A022916.
%Y 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.
%Y See A268240 for this read mod 2.
%Y Cf. A013609 (row sums).
%K nonn,tabf,look,easy
%O 0,6
%A _Lior Manor_
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