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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 28 2019
STATUS
approved