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A137448 Triangle T(n,k) = (1-k*(k-1))*A053120(n,k), read by rows, 0<=k<=n.

%I A137448 #12 Mar 17 2023 02:11:43
%S A137448 1,0,1,-1,0,-2,0,-3,0,-20,1,0,8,0,-88,0,5,0,100,0,-304,-1,0,-18,0,528,
%T A137448 0,-928,0,-7,0,-280,0,2128,0,-2624,1,0,32,0,-1760,0,7424,0,-7040,0,9,
%U A137448 0,600,0,-8208,0,23616,0,-18176,-1,0,-50,0,4400,0,-32480,0,70400,0,-45568
%N A137448 Triangle T(n,k) = (1-k*(k-1))*A053120(n,k), read by rows, 0<=k<=n.
%C A137448 The entries may also be defined as the coefficient [x^k] of the polynomial T(n,x)-x^2* (d^2/dx^2) T(n,x) where T are the Chebyshev polynomials (A053120).
%C A137448 Row sums are 1, 1, -3, -23, -79, -199, -419, -783, -1343, -2159, -3299, ...
%F A137448 Row sums: Sum_{k=0..n} T(n,k) = (3+n^2-n^4)/3 = 1-A112742(n). - _R. J. Mathar_, Sep 10 2013
%e A137448 1;
%e A137448 0, 1;
%e A137448 -1, 0, -2;
%e A137448 0, -3, 0, -20;
%e A137448 1, 0, 8, 0, -88;
%e A137448 0, 5, 0, 100, 0, -304;
%e A137448 -1, 0, -18, 0, 528, 0, -928;
%e A137448 0, -7, 0, -280, 0, 2128, 0, -2624;
%e A137448 1, 0, 32, 0, -1760, 0, 7424, 0, -7040;
%e A137448 0, 9, 0, 600, 0, -8208,0, 23616, 0, -18176;
%e A137448 -1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568;
%t A137448 Clear[p, x, a] p[x, 0] = 1; p[x, 1] = x + 1; p[x_, n_] := p[x, n] = -x^2*D[ChebyshevT[n, x], {x, 2}] + ChebyshevT[n, x]; Table[Expand[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]
%Y A137448 Cf. A053120, A112742.
%K A137448 tabl,sign
%O A137448 0,6
%A A137448 _Roger L. Bagula_ and _Gary W. Adamson_, Apr 18 2008
%E A137448 T(1,0) corrected by _R. J. Mathar_, Sep 10 2013