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A084600
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Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x+2x^2)^n for n >= 0.
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17
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1, 1, 1, 2, 1, 2, 5, 4, 4, 1, 3, 9, 13, 18, 12, 8, 1, 4, 14, 28, 49, 56, 56, 32, 16, 1, 5, 20, 50, 105, 161, 210, 200, 160, 80, 32, 1, 6, 27, 80, 195, 366, 581, 732, 780, 640, 432, 192, 64, 1, 7, 35, 119, 329, 721, 1337, 2045, 2674, 2884, 2632, 1904, 1120, 448, 128, 1, 8, 44
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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Triangle by rows, X^n * [1,0,0,0,...]; where X = an infinite tridiagonal matrix with (1,1,1,...) in the main and subdiagonals and (2,2,2,...) in the subsubdiagonal. Also, X = an infinite triangular matrix with (1,1,2,0,0,0,...) in each column. - Gary W. Adamson, May 26 2008
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LINKS
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FORMULA
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G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x^(2*k+1)*(1+x+2*x^2)/(x^(2*k+1)*(1+x+2*x^2) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
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EXAMPLE
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Triangle begins:
1;
1, 1, 2;
1, 2, 5, 4, 4;
1, 3, 9, 13, 18, 12, 8;
1, 4, 14, 28, 49, 56, 56, 32, 16;
1, 5, 20, 50, 105, 161, 210, 200, 160, 80, 32;
1, 6, 27, 80, 195, 366, 581, 732, 780, 640, 432, 192, 64;
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MAPLE
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f:= proc(n) option remember; expand((1+x+2*x^2)^n) end:
T:= (n, k)-> coeff(f(n), x, k):
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MATHEMATICA
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t[n_, k_] := Coefficient[(1+x+2x^2)^n, x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, 2 n}] // Flatten (* Jean-François Alcover, Feb 27 2015 *)
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PROG
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(Haskell)
a084600 n = a084600_list !! n
a084600_list = concat $ iterate ([1, 1, 2] *) [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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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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