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A084600 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x+2x^2)^n for n>=0. 17
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)
OFFSET

0,4

COMMENTS

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

Row sums = (1, 4, 16, 64, 256,...). - Gary W. Adamson, May 26 2008

LINKS

Alois P. Heinz, Rows n = 0..100, flattened

G. Farkas, G. Kallos and G. Kiss, Large primes in generalized Pascal triangles, Acta Univ. Sapientiae, Informatica, 3, 2 (2011) 158-171.

FORMULA

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

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,

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

MATHEMATICA

t[n_, k_] := Coefficient[(1+x+2x^2)^n, x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, 2 n}] // Flatten (* Jean-Fran├žois Alcover, Feb 27 2015 *)

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

CROSSREFS

Cf. A002426, A084601-A084615.

Sequence in context: A078016 A078046 A319200 * A216760 A318724 A167482

Adjacent sequences:  A084597 A084598 A084599 * A084601 A084602 A084603

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Jun 01 2003

STATUS

approved

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Last modified August 22 11:34 EDT 2019. Contains 326176 sequences. (Running on oeis4.)