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A329918
Coefficients of orthogonal polynomials related to the Jacobsthal numbers A152046, triangle read by rows, T(n, k) for 0 <= k <= n.
1
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
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.
Sequence in context: A036852 A260941 A352996 * A281442 A256038 A050327
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 28 2019
STATUS
approved