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%I #8 Aug 06 2023 08:17:34
%S 1,-2,1,7,-6,1,-36,40,-12,1,253,-326,131,-20,1,-2278,3233,-1552,324,
%T -30,1,25059,-38140,20678,-5260,675,-42,1,-325768,523456,-310560,
%U 90754,-14380,1252,-56,1,4886521,-8205244,5223602,-1694244,312059,-33866,2135,-72,1
%N Coefficient array of orthogonal polynomials P(n,x)=(x-2n)*P(n-1,x)-(2n-3)*P(n-2,x), P(0,x)=1,P(1,x)=x-2.
%C Inverse is A178121. First column is A112293 signed.
%e Triangle begins
%e 1,
%e -2, 1,
%e 7, -6, 1,
%e -36, 40, -12, 1,
%e 253, -326, 131, -20, 1,
%e -2278, 3233, -1552, 324, -30, 1,
%e 25059, -38140, 20678, -5260, 675, -42, 1,
%e -325768, 523456, -310560, 90754, -14380, 1252, -56, 1,
%e 4886521, -8205244, 5223602, -1694244, 312059, -33866, 2135, -72, 1
%e Production matrix of inverse is
%e 2, 1,
%e 1, 4, 1,
%e 0, 3, 6, 1,
%e 0, 0, 5, 8, 1,
%e 0, 0, 0, 7, 10, 1,
%e 0, 0, 0, 0, 9, 12, 1,
%e 0, 0, 0, 0, 0, 11, 14, 1,
%e 0, 0, 0, 0, 0, 0, 13, 16, 1,
%e 0, 0, 0, 0, 0, 0, 0, 15, 18, 1
%p A178120 := proc(n,k)
%p if n = k then
%p 1;
%p elif n = 1 and k = 0 then
%p -2 ;
%p elif k < 0 or k > n then
%p 0 ;
%p else
%p -2*n*procname(n-1,k)+procname(n-1,k-1)-(2*n-3)*procname(n-2,k) ;
%p end if;
%p end proc: # _R. J. Mathar_, Dec 03 2014
%t P[0, _] = 1;
%t P[1, x_] := x - 2;
%t P[n_, x_] := P[n, x] = (x-2n) P[n-1, x] - (2n-3) P[n-2, x];
%t T[n_] := Module[{x}, CoefficientList[P[n, x], x]];
%t Table[T[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Aug 06 2023 *)
%K sign,easy,tabl
%O 0,2
%A _Paul Barry_, May 20 2010
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