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A329918
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Coefficients of orthogonal polynomials related to the Jacobsthal numbers A152046, triangle read by rows, T(n, k) for 0 <= k <= n.
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1
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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
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OFFSET
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0,8
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LINKS
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FORMULA
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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.
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EXAMPLE
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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;
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MAPLE
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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);
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PROG
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(Julia)
using Nemo # Returns row n.
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
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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