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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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16
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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; internal format)
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OFFSET
| 0,4
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COMMENTS
| Triangl 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 (qntmpkt(AT)yahoo.com), May 26 2008
Row sums = (1, 4, 16, 64, 256,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 26 2008
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REFERENCES
| G. Farkas, G. Kallos and G. Kiss, Large primes in generalized Pascal triangles Acta Univ. Sapientiae, Informatica, 3, 2 (2011) 158-171; http://www.acta.sapientia.ro/acta-info/C3-2/info32-2.pdf.
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EXAMPLE
| 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
| f:= proc(n) option remember; expand ((1+x+2*x^2)^n) end:
T:= (n, k)-> coeff (f(n), x, k):
seq (seq (T(n, k), k=0..2*n), n=0..10); # Alois P. Heinz, Apr 03 2011
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PROG
| (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
_ * _ = []
-- Reinhard Zumkeller, Apr 02 2011
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CROSSREFS
| Cf. A002426, A084601-A084615.
Sequence in context: A136457 A078016 A078046 * A167482 A137327 A143913
Adjacent sequences: A084597 A084598 A084599 * A084601 A084602 A084603
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KEYWORD
| nonn,tabf
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 01 2003
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