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A209135 Triangle of coefficients of polynomials u(n,x) jointly generated with A209136; see the Formula section.

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