A329918 Coefficients of orthogonal polynomials related to the Jacobsthal numbers A152046, triangle read by rows, T(n, k) for 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 4, 0, 1, 0, 4, 0, 6, 0, 1, 0, 0, 12, 0, 8, 0, 1, 0, 8, 0, 24, 0, 10, 0, 1, 0, 0, 32, 0, 40, 0, 12, 0, 1, 0, 16, 0, 80, 0, 60, 0, 14, 0, 1, 0, 0, 80, 0, 160, 0, 84, 0, 16, 0, 1, 0, 32, 0, 240, 0, 280, 0, 112, 0, 18, 0, 1

Offset

0,8

Links

Table of n, a(n) for n=0..77.

Formula

p(n) = x*p(n-1) + 2*p(n-2) for n >= 3; p(0) = 1, p(1) = x, p(2) = x^2.

T(n, k) = [x^k] p(n).

T(n, k) = 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2)) if n+k is even otherwise 0.

Example

Triangle starts:
[0] 1;
[1] 0,  1;
[2] 0,  0,  1;
[3] 0,  2,  0,  1;
[4] 0,  0,  4,  0,  1;
[5] 0,  4,  0,  6,  0,  1;
[6] 0,  0, 12,  0,  8,  0,  1;
[7] 0,  8,  0, 24,  0, 10,  0,  1;
[8] 0,  0, 32,  0, 40,  0, 12,  0, 1;
[9] 0, 16,  0, 80,  0, 60,  0, 14, 0, 1;
The first few polynomials:
p(0,x) = 1;
p(1,x) = x;
p(2,x) = x^2;
p(3,x) = 2*x + x^3;
p(4,x) = 4*x^2 + x^4;
p(5,x) = 4*x + 6*x^3 + x^5;
p(6,x) = 12*x^2 + 8*x^4 + x^6;

Maple

T := (n, k) -> `if`((n+k)::odd, 0, 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2)):
seq(seq(T(n, k), k=0..n), n=0..11);

Prog

(Julia)
using Nemo # Returns row n.
function A329918(row)
    R, x = PolynomialRing(ZZ, "x")
    function p(n)
        n < 3 && return x^n
        x*p(n-1) + 2*p(n-2)
    end
    p = p(row)
    [coeff(p, k) for k in 0:row]
end
for row in 0:9 println(A329918(row)) end # prints triangle

Crossrefs

Row sums are A001045 starting with 1, which is A152046. These are in signed form also the alternating row sums. Diagonal sums are aerated A133494.

Cf. A110509, A113953, A114192, A167431, A322942.

Sequence in context: A036852 A260941 A352996 * A281442 A256038 A050327

Adjacent sequences: A329915 A329916 A329917 * A329919 A329920 A329921

Keyword

nonn,tabl

Author

Peter Luschny, Nov 28 2019

Status

approved