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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
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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}).

Row sums are tribonacci numbers.

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 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".

FORMULA

G.f.: x/(1-x-x^2*y-x^3*y^2). - Vladeta Jovovic, May 30 2003

From Werner Schulte, Feb 22 2017: (Start)

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).

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.

(End)

EXAMPLE

Triangle begins:

  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;

MAPLE

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);

MATHEMATICA

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 *)

PROG

(Haskell)

a082601 n k = a082601_tabl !! n !! k

a082601_row n = a082601_tabl !! n

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

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

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

-- Reinhard Zumkeller, Apr 13 2014

CROSSREFS

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

Cf. A002426 (central terms).

Sequence in context: A173438 A103493 A121480 * A286509 A213887 A279589

Adjacent sequences:  A082598 A082599 A082600 * A082602 A082603 A082604

KEYWORD

nonn,tabl,easy

AUTHOR

Gary W. Adamson, May 24 2003

EXTENSIONS

Edited by Anne Donovan and N. J. A. Sloane, May 27 2003

More terms from Emeric Deutsch, May 06 2004

STATUS

approved

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Last modified February 22 13:09 EST 2018. Contains 299454 sequences. (Running on oeis4.)