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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.

%I A329918 #22 Dec 18 2024 22:14:00
%S A329918 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 A329918 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 A329918 84,0,16,0,1,0,32,0,240,0,280,0,112,0,18,0,1
%N A329918 Coefficients of orthogonal polynomials related to the Jacobsthal numbers A152046, triangle read by rows, T(n, k) for 0 <= k <= n.
%F A329918 p(n) = x*p(n-1) + 2*p(n-2) for n >= 3; p(0) = 1, p(1) = x, p(2) = x^2.
%F A329918 T(n, k) = [x^k] p(n).
%F A329918 T(n, k) = 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2) if n+k is even otherwise 0.
%e A329918 Triangle starts:
%e A329918   [0] 1;
%e A329918   [1] 0,  1;
%e A329918   [2] 0,  0,  1;
%e A329918   [3] 0,  2,  0,  1;
%e A329918   [4] 0,  0,  4,  0,  1;
%e A329918   [5] 0,  4,  0,  6,  0,  1;
%e A329918   [6] 0,  0, 12,  0,  8,  0,  1;
%e A329918   [7] 0,  8,  0, 24,  0, 10,  0,  1;
%e A329918   [8] 0,  0, 32,  0, 40,  0, 12,  0, 1;
%e A329918   [9] 0, 16,  0, 80,  0, 60,  0, 14, 0, 1;
%e A329918 The first few polynomials:
%e A329918   p(0,x) = 1;
%e A329918   p(1,x) = x;
%e A329918   p(2,x) = x^2;
%e A329918   p(3,x) = 2*x + x^3;
%e A329918   p(4,x) = 4*x^2 + x^4;
%e A329918   p(5,x) = 4*x + 6*x^3 + x^5;
%e A329918   p(6,x) = 12*x^2 + 8*x^4 + x^6;
%p A329918 T := (n, k) -> `if`((n+k)::odd, 0, 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2)):
%p A329918 seq(seq(T(n, k), k=0..n), n=0..11);
%o A329918 (Julia)
%o A329918 using Nemo # Returns row n.
%o A329918 function A329918(row)
%o A329918     R, x = PolynomialRing(ZZ, "x")
%o A329918     function p(n)
%o A329918         n < 3 && return x^n
%o A329918         x*p(n-1) + 2*p(n-2)
%o A329918     end
%o A329918     p = p(row)
%o A329918     [coeff(p, k) for k in 0:row]
%o A329918 end
%o A329918 for row in 0:9 println(A329918(row)) end # prints triangle
%Y A329918 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 A329918 Cf. A110509, A113953, A114192, A167431, A322942.
%K A329918 nonn,tabl
%O A329918 0,8
%A A329918 _Peter Luschny_, Nov 28 2019