%I #6 Dec 26 2023 15:34:43
%S 1,0,1,-1,0,1,0,-2,0,1,3,0,-5,0,1,0,9,0,-8,0,1,-15,0,34,0,-13,0,1,0,
%T -60,0,74,0,-18,0,1,105,0,-298,0,165,0,-25,0,1,0,525,0,-816,0,291,0,
%U -32,0,1,-945,0,3207,0,-2301,0,516,0,-41,0,1,0,-5670,0,10551,0,-4920,0,804,0
%N Coefficient array for orthogonal polynomials P(n,x)=x*P(n-1,x)-(2*floor((n+2)/2)-3)*P(n-2,x), P(0,x)=1,P(1,x)=x.
%C Inverse is A178108. First column is signed aerated version of double factorials A001147.
%e Triangle begins
%e 1,
%e 0, 1,
%e -1, 0, 1,
%e 0, -2, 0, 1,
%e 3, 0, -5, 0, 1,
%e 0, 9, 0, -8, 0, 1,
%e -15, 0, 34, 0, -13, 0, 1,
%e 0, -60, 0, 74, 0, -18, 0, 1,
%e 105, 0, -298, 0, 165, 0, -25, 0, 1,
%e 0, 525, 0, -816, 0, 291, 0, -32, 0, 1,
%e -945, 0, 3207, 0, -2301, 0, 516, 0, -41, 0, 1
%e Production matrix is
%e 0, 1,
%e -1, 0, 1,
%e 0, -1, 0, 1,
%e 1, 0, -3, 0, 1,
%e 0, 1, 0, -3, 0, 1,
%e 2, 0, 1, 0, -5, 0, 1,
%e 0, 2, 0, 1, 0, -5, 0, 1,
%e 7, 0, 2, 0, 1, 0, -7, 0, 1,
%e 0, 7, 0, 2, 0, 1, 0, -7, 0, 1
%e Production matrix of inverse is
%e 0, 1,
%e 1, 0, 1,
%e 0, 1, 0, 1,
%e 0, 0, 3, 0, 1,
%e 0, 0, 0, 3, 0, 1,
%e 0, 0, 0, 0, 5, 0, 1,
%e 0, 0, 0, 0, 0, 5, 0, 1,
%e 0, 0, 0, 0, 0, 0, 7, 0, 1,
%e 0, 0, 0, 0, 0, 0, 0, 7, 0, 1
%t p[0] = 1; p[1] = x;
%t p[n_] := p[n] = Expand[x p[n-1] - (2 Quotient[n+2, 2] - 3) p[n-2]];
%t Table[CoefficientList[p[n], x], {n, 0, 6}] (* _Andrey Zabolotskiy_, Dec 26 2023 *)
%K sign,tabl
%O 0,8
%A _Paul Barry_, May 20 2010