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 A082601 Tribonacci array: to get next row, right-adjust previous 3 rows and add them, then append a final 0. 5

%I

%S 1,1,0,1,1,0,1,2,1,0,1,3,3,0,0,1,4,6,2,0,0,1,5,10,7,1,0,0,1,6,15,16,6,

%T 0,0,0,1,7,21,30,19,3,0,0,0,1,8,28,50,45,16,1,0,0,0,1,9,36,77,90,51,

%U 10,0,0,0,0,1,10,45,112,161,126,45,4,0,0,0,0,1,11,55,156,266,266,141,30,1,0

%N Tribonacci array: to get next row, right-adjust previous 3 rows and add them, then append a final 0.

%C Coefficients of tribonacci polynomials: t_0 = 1, t_1 = x, t_2 = x^2+x, t_n = x*(t_{n-1}+t_{n-2}+t_{n-3}).

%C Row sums are tribonacci numbers.

%H Reinhard Zumkeller, <a href="/A082601/b082601.txt">Rows n = 0..125 of triangle, flattened</a>

%H Thomas Koshy, <a href="https://dx.doi.org/10.1002/9781118033067">Fibonacci and Lucas Numbers with Applications</a>, Wiley, 2001; Chapter 47: Tribonacci Polynomials: ("In 1973, V.E. Hoggatt, Jr. and M. Bicknell generalized Fibonacci polynomials to Tribonacci polynomials tx(x)"); Table 47.1, page 534: "Tribonacci Array".

%F G.f.: x/(1-x-x^2*y-x^3*y^2). - _Vladeta Jovovic_, May 30 2003

%F From _Werner Schulte_, Feb 22 2017: (Start)

%F T(n,k) = Sum_{j=0..floor(k/2)} binomial(k-j,j)*binomial(n-k,k-j) for 0<=k and k<=floor(2*n/3) with binomial(i,j)=0 for i<j (see _Dennis P. Walsh_ at A078802).

%F Based on two integers p and q define the integer sequence U(n) by U(0)=0 and U(1)=0 and U(n+2) = Sum_{k=0..floor(2*n/3)} T(n,k)*p^k*q^(2*n-3*k) for n>=0. That yields the g.f. f(p,q,x) = x^2/(1-q^2*x-p*q*x^2-p^2*x^3) and the recurrence U(n+3) = q^2*U(n+2)+p*q*U(n+1)+p^2*U(n) for n>=0 with initial values U(0)=U(1)=0 and U(2)=1. For p = q = +-1 you'll get tribonacci numbers A000073. For p = -1 and q = 1 you'll get A021913.

%F (End)

%e Triangle begins:

%e 1;

%e 1, 0;

%e 1, 1, 0;

%e 1, 2, 1, 0;

%e 1, 3, 3, 0, 0;

%e 1, 4, 6, 2, 0, 0;

%e 1, 5, 10, 7, 1, 0, 0;

%p G:=x*y/(1-x-x^2*y-x^3*y^2): Gs:=simplify(series(G,x=0,18)): for n from 1 to 16 do P[n]:=sort(coeff(Gs,x^n)) od: seq(seq(coeff(P[i],y^j),j=1..i),i=1..16);

%t Table[SeriesCoefficient[x/(1 - x - x^2*y - x^3*y^2), {x, 0, n}, {y, 0, k}], {n, 13}, {k, 0, n - 1}] // Flatten (* _Michael De Vlieger_, Feb 22 2017 *)

%o a082601 n k = a082601_tabl !! n !! k

%o a082601_row n = a082601_tabl !! n

%o a082601_tabl = [1] : [1,0] : [1,1,0] : f [0,0,1] [0,1,0] [1,1,0]

%o where f us vs ws = ys : f (0:vs) (0:ws) ys where

%o ys = zipWith3 (((+) .) . (+)) us vs ws ++ [0]

%o -- _Reinhard Zumkeller_, Apr 13 2014

%Y Closely related to A078802. A better version of A082870. Cf. A000073.

%Y Cf. A002426 (central terms).

%K nonn,tabl,easy

%O 0,8

%A _Gary W. Adamson_, May 24 2003

%E Edited by Anne Donovan and _N. J. A. Sloane_, May 27 2003

%E More terms from _Emeric Deutsch_, May 06 2004

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Last modified February 18 20:04 EST 2019. Contains 320262 sequences. (Running on oeis4.)