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%I #31 Feb 06 2022 06:56:09
%S 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,
%T 50,105,161,210,200,160,80,32,1,6,27,80,195,366,581,732,780,640,432,
%U 192,64,1,7,35,119,329,721,1337,2045,2674,2884,2632,1904,1120,448,128,1,8,44
%N Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x+2x^2)^n for n >= 0.
%C 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
%C Row sums = (1, 4, 16, 64, 256, ...). - _Gary W. Adamson_, May 26 2008
%H Alois P. Heinz, <a href="/A084600/b084600.txt">Rows n = 0..100, flattened</a>
%H G. Farkas, G. Kallos and G. Kiss, <a href="http://www.acta.sapientia.ro/acta-info/C3-2/info32-2.pdf">Large primes in generalized Pascal triangles</a>, Acta Univ. Sapientiae, Informatica, 3, 2 (2011) 158-171.
%F 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
%e Triangle begins:
%e 1;
%e 1, 1, 2;
%e 1, 2, 5, 4, 4;
%e 1, 3, 9, 13, 18, 12, 8;
%e 1, 4, 14, 28, 49, 56, 56, 32, 16;
%e 1, 5, 20, 50, 105, 161, 210, 200, 160, 80, 32;
%e 1, 6, 27, 80, 195, 366, 581, 732, 780, 640, 432, 192, 64;
%p f:= proc(n) option remember; expand((1+x+2*x^2)^n) end:
%p T:= (n,k)-> coeff(f(n), x, k):
%p seq(seq(T(n, k), k=0..2*n), n=0..10); # _Alois P. Heinz_, Apr 03 2011
%t 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 *)
%o (Haskell)
%o a084600 n = a084600_list !! n
%o a084600_list = concat $ iterate ([1,1,2] *) [1]
%o instance Num a => Num [a] where
%o fromInteger k = [fromInteger k]
%o (p:ps) + (q:qs) = p + q : ps + qs
%o ps + qs = ps ++ qs
%o (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
%o _ * _ = []
%o -- _Reinhard Zumkeller_, Apr 02 2011
%Y Cf. A002426, A084601-A084615.
%K nonn,tabf
%O 0,4
%A _Paul D. Hanna_, Jun 01 2003
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