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
Examples
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; ...
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.
Crossrefs
p_n(3) gives A005013.
Programs
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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 *)
Extensions
More terms from Alois P. Heinz, Jul 10 2015