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A208757 Triangle of coefficients of polynomials u(n,x) jointly generated with A208758; see the Formula section.

%I A208757 #14 Jan 22 2020 20:13:30
%S A208757 1,1,2,1,2,6,1,2,8,16,1,2,10,24,44,1,2,12,32,76,120,1,2,14,40,112,232,
%T A208757 328,1,2,16,48,152,368,704,896,1,2,18,56,196,528,1200,2112,2448,1,2,
%U A208757 20,64,244,712,1824,3840,6288,6688,1,2,22,72,296,920,2584,6144
%N A208757 Triangle of coefficients of polynomials u(n,x) jointly generated with A208758; see the Formula section.
%C A208757 For a discussion and guide to related arrays, see A208510.
%C A208757 Subtriangle of the triangle (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 18 2012
%F A208757 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208757 v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
%F A208757 where u(1,x)=1, v(1,x)=1.
%F A208757 From _Philippe Deléham_, Mar 18 2012: (Start)
%F A208757 As DELTA-triangle with 0 <= k <= n:
%F A208757 G.f.: (1-2*y*x+2*y*x^2-2*y^2*x^2)/(1-x-2*y*x+2*y*x^2-2*y^2*x^2).
%F A208757 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) -2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A208757 First five rows:
%e A208757   1;
%e A208757   1, 2;
%e A208757   1, 2,  6;
%e A208757   1, 2,  8, 16;
%e A208757   1, 2, 10, 24, 44;
%e A208757 First five polynomials u(n,x):
%e A208757   1
%e A208757   1 + 2x
%e A208757   1 + 2x +  6x^2
%e A208757   1 + 2x +  8x^2 + 16x^3
%e A208757   1 + 2x + 10x^2 + 24x^3 + 44x^4
%e A208757 From _Philippe Deléham_, Mar 18 2012: (Start)
%e A208757 (1, 0, -1, 1, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, ...) begins:
%e A208757   1
%e A208757   1, 0
%e A208757   1, 2,  0
%e A208757   1, 2,  6,  0
%e A208757   1, 2,  8, 16,   0
%e A208757   1, 2, 10, 24,  44,   0
%e A208757   1, 2, 12, 32,  76, 120,   0
%e A208757   1, 2, 14, 40, 112, 232, 328, 0; (End)
%t A208757 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208757 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208757 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
%t A208757 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208757 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208757 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208757 TableForm[cu]
%t A208757 Flatten[%]  (* A208757 *)
%t A208757 Table[Expand[v[n, x]], {n, 1, z}]
%t A208757 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208757 TableForm[cv]
%t A208757 Flatten[%]  (* A208758 *)
%Y A208757 Cf. A208758, A208510.
%K A208757 nonn,tabl
%O A208757 1,3
%A A208757 _Clark Kimberling_, Mar 02 2012