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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: a(i+1, k, j) = a(i, k, j)+a(i, j, k-1)+a(i, j-1, k-1), a(i, j, -1) := 0, ...; a(0, 0, 0)=1.

T(n,j,k) = n!/(j!*k!*(n-i-j)!). - Gregory Gerard Wojnar, Sep 30 2018

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-Fran├žois 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

Lior Manor

STATUS

approved

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Last modified December 12 07:31 EST 2019. Contains 329948 sequences. (Running on oeis4.)