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

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

Offset: 0

Author

Paul D. Hanna, Jun 01 2003

Keywords

• nonn
• tabf

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

Examples

			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;
		

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.

Crossrefs

Cf. A002426, A084601-A084615.

Programs

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

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-François Alcover, Feb 27 2015 *)

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