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%I #13 Jan 24 2020 03:25:45
%S 1,2,1,3,5,3,4,14,16,5,5,30,54,39,11,6,55,144,171,98,21,7,91,329,561,
%T 503,229,43,8,140,672,1534,1928,1380,532,85,9,204,1260,3690,6106,6084,
%U 3636,1203,171,10,285,2208,8058,16852,21890,18060,9249,2694
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209136; see the Formula section.
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 11 2012
%F u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Apr 11 2012: (Start)
%F As DELTA-triangle T(n,k) with 0 <= k <= n:
%F G.f.: (1-x-y*x+x^2-y*x^2-2*y^2*x^2)/(1-2*x-y*x+x^2-y*x^2-2*y^2*x^2).
%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e First five rows:
%e 1;
%e 2, 1;
%e 3, 5, 3;
%e 4, 14, 16, 5;
%e 5, 30, 54, 39, 11;
%e First three polynomials u(n,x):
%e 1
%e 2 + x
%e 3 + 5x + 3x^2
%e From _Philippe Deléham_, Apr 11 2012: (Start)
%e (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 2, 1, 0;
%e 3, 5, 3, 0;
%e 4, 14, 16, 5, 0;
%e 5, 30, 54, 39, 11, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A209135 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A209136 *)
%Y Cf. A209136, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 05 2012
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