A329918 - OEIS

%I #19 Nov 30 2019 09:20:27

%S 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,

%T 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,

%U 84,0,16,0,1,0,32,0,240,0,280,0,112,0,18,0,1

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

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

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

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

%e Triangle starts:

%e [0] 1;

%e [1] 0,  1;

%e [2] 0,  0,  1;

%e [3] 0,  2,  0,  1;

%e [4] 0,  0,  4,  0,  1;

%e [5] 0,  4,  0,  6,  0,  1;

%e [6] 0,  0, 12,  0,  8,  0,  1;

%e [7] 0,  8,  0, 24,  0, 10,  0,  1;

%e [8] 0,  0, 32,  0, 40,  0, 12,  0, 1;

%e [9] 0, 16,  0, 80,  0, 60,  0, 14, 0, 1;

%e The first few polynomials:

%e p(0,x) = 1;

%e p(1,x) = x;

%e p(2,x) = x^2;

%e p(3,x) = 2*x + x^3;

%e p(4,x) = 4*x^2 + x^4;

%e p(5,x) = 4*x + 6*x^3 + x^5;

%e p(6,x) = 12*x^2 + 8*x^4 + x^6;

%p T := (n, k) -> `if`((n+k)::odd, 0, 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2)):

%p seq(seq(T(n, k), k=0..n), n=0..11);

%o (Julia)

%o using Nemo # Returns row n.

%o function A329918(row)

%o     R, x = PolynomialRing(ZZ, "x")

%o     function p(n)

%o         n < 3 && return x^n

%o         x*p(n-1) + 2*p(n-2)

%o     end

%o     p = p(row)

%o     [coeff(p, k) for k in 0:row]

%o end

%o for row in 0:9 println(A329918(row)) end # prints triangle

%Y 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.

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

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Nov 28 2019