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A046816
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Entries in 3-dimensional version of Pascal triangle: trinomial coefficients.
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17
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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
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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
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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 [From Vincenzo Librandi, Jul 19 2009]
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..10659
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FORMULA
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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.
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EXAMPLE
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... 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);
# Alois P. Heinz, Apr 03 2011
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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 *)
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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
_ * _ = []
-- Reinhard Zumkeller, Apr 02 2011
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CROSSREFS
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Cf. A007318.
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.
Sequence in context: A089955 A180312 A178819 * A138328 A137264 A193238
Adjacent sequences: A046813 A046814 A046815 * A046817 A046818 A046819
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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Lior Manor
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STATUS
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approved
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