A259875 Irregular triangle read by rows: coefficients (highest degree first) of polynomials defined by p_0(x)=0, p_1(x)=p_2(x)=1, p_3(x)=x+1; p_n(x)=x*p_{n-2}(x)-p_{n-4}(x).

0, 1, 1, 1, 1, 1, 0, 1, 1, -1, 1, 0, -1, 1, 1, -2, -1, 1, 0, -2, 0, 1, 1, -3, -2, 1, 1, 0, -3, 0, 1, 1, 1, -4, -3, 3, 1, 1, 0, -4, 0, 3, 0, 1, 1, -5, -4, 6, 3, -1, 1, 0, -5, 0, 6, 0, -1, 1, 1, -6, -5, 10, 6, -4, -1, 1, 0, -6, 0, 10, 0, -4, 0, 1, 1, -7, -6, 15, 10, -10, -4, 1

Offset

0,16

Links

Alois P. Heinz, Rows n = 0..200, flattened

A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979. [Annotated scanned copy]

A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.

Example

Triangle begins:
0;
1;
1;
1, 1;
1, 0;
1, 1, -1;
1, 0, -1;
1, 1, -2, -1;
1, 0, -2,  0;
1, 1, -3, -2, 1;
1, 0, -3,  0, 1;
1, 1, -4, -3, 3, 1;
1, 0, -4,  0, 3, 0;
...

Maple

p:= proc(n) option remember; expand(`if`(n=0, 0,
     `if`(n<3, 1, `if`(n=3, x+1, x*p(n-2)-p(n-4)))))
    end:
T:= n-> `if`(n=0, 0, (s-> seq(coeff(s, x, degree(s)-i)
        , i=0..degree(s)))(p(n))):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 10 2015

Mathematica

p[0] = 0&; p[1] = p[2] = 1&; p[3] = #+1&; p[n_][x_] := p[n, x] = x*p[n-2][x] - p[n-4][x];
row[0] = {0}; row[n_] := CoefficientList[p[n][x], x] // Reverse;
Table[row[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jun 12 2018 *)

Crossrefs

p_n(3) gives A005013.

Sequence in context: A136167 A140748 A185305 * A070821 A165890 A051632

Adjacent sequences: A259872 A259873 A259874 * A259876 A259877 A259878

Keyword

sign,tabf,look

Author

N. J. A. Sloane, Jul 09 2015

Extensions

More terms from Alois P. Heinz, Jul 10 2015

Status

approved