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A137350 A triangular Sequence of coefficients of a three deep polynomial recursion based on a Chebyshev kind and a Padovan recursion: Chebyshev; p(x,n)=x*p(x,n-1)-p(x,n-2); Padovan: a(n)=a(n-2)+a(n-3); Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).

%I A137350 #3 Mar 30 2012 17:34:26
%S A137350 1,-1,1,0,1,-1,-1,1,1,-1,1,0,-2,-1,1,1,2,-2,1,-1,1,-3,-1,1,0,3,3,-3,1,
%T A137350 -1,-3,3,-4,-1,1,1,-1,6,4,-4,1
%N A137350 A triangular Sequence of coefficients of a three deep polynomial recursion based on a Chebyshev kind and a Padovan recursion: Chebyshev; p(x,n)=x*p(x,n-1)-p(x,n-2); Padovan: a(n)=a(n-2)+a(n-3); Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).
%C A137350 Row sums are:
%C A137350 {1, 0, 1, -1, 1, -2, 2, -3, 4, -5, 7}
%C A137350 In differential equation terms this is equivalent to ( in Mathematica notation):
%C A137350 D[y[x],{x,3}]=x*D[y[x],{x,1}]-y[x];
%C A137350 Two simple possible HypergeometricPFQ based results are:
%C A137350 DSolve[{D[y[x], {x, 3}] == x*D[y[x], {x, 1}] - y[x], y[0] == 1}, y, x];
%C A137350 DSolve[{D[y[x], {x, 3}] == x*D[y[x], {x, 1}] - y[x], y[0] == 0}, y, x].
%F A137350 Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).
%e A137350 {1},
%e A137350 {-1, 1},
%e A137350 {0, 1},
%e A137350 {-1, -1, 1},
%e A137350 {1, -1, 1},
%e A137350 {0, -2, -1, 1},
%e A137350 {1, 2, -2, 1},
%e A137350 {-1, 1, -3, -1, 1},
%e A137350 {0, 3, 3, -3, 1},
%e A137350 {-1, -3, 3, -4, -1, 1},
%e A137350 {1, -1, 6, 4, -4, 1}
%t A137350 Clear[Q, x] Q[x, -2] = 1 - x; Q[x, -1] = 0; Q[x, 0] = 1; Q[x_, n_] := Q[x, n] = x*Q[x, n - 2] - Q[x, n - 3]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]
%Y A137350 Cf. A000931, A137276.
%K A137350 uned,tabl,sign
%O A137350 1,13
%A A137350 _Roger L. Bagula_, Apr 08 2008