login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)