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 A046816 Pascal's tetrahedron: entries in 3-dimensional version of Pascal triangle, or trinomial coefficients. 23
 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)
 OFFSET 0,6 COMMENTS 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 REFERENCES 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] LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10659 Wikipedia, Pascal's pyramid FORMULA 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 G.f.: 1/(1-x-x*y-x*y*z). - Georg Fischer, May 29 2019 EXAMPLE 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 ---------------- ... MAPLE 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); #  Alois P. Heinz, Apr 03 2011 MATHEMATICA 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 *) PROG (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    _ * _               = [] -- Reinhard Zumkeller, Apr 02 2011 CROSSREFS Cf. A007318, A022916. 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. See A268240 for this read mod 2. Sequence in context: A089955 A180312 A178819 * A301475 A138328 A137264 Adjacent sequences:  A046813 A046814 A046815 * A046817 A046818 A046819 KEYWORD nonn,tabf,look,easy AUTHOR STATUS approved

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Last modified May 11 00:43 EDT 2021. Contains 343784 sequences. (Running on oeis4.)