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A208751 Triangle of coefficients of polynomials u(n,x) jointly generated with A208752; see the Formula section.

%I A208751 #14 Jan 24 2020 03:30:02
%S A208751 1,1,2,1,6,2,1,12,12,2,1,20,40,18,2,1,30,100,86,24,2,1,42,210,294,150,
%T A208751 30,2,1,56,392,812,656,232,36,2,1,72,672,1932,2268,1240,332,42,2,1,90,
%U A208751 1080,4116,6624,5172,2100,450,48,2,1,110,1650,8052,17028,17996
%N A208751 Triangle of coefficients of polynomials u(n,x) jointly generated with A208752; see the Formula section.
%C A208751 For a discussion and guide to related arrays, see A208510.
%C A208751 Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 17 2012
%F A208751 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208751 v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x),
%F A208751 where u(1,x)=1, v(1,x)=1.
%F A208751 From _Philippe Deléham_, Mar 17 2012: (Start)
%F A208751 As DELTA-triangle with 0 <= k <= n:
%F A208751 G.f.: (1-x-y*x)/(1-2*x-y*x+x^2-y*x^2).
%F A208751 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1), 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 A208751 First five rows:
%e A208751   1;
%e A208751   1,  2;
%e A208751   1,  6,  2;
%e A208751   1, 12, 12,  2;
%e A208751   1, 20, 40, 18,  2;
%e A208751 First five polynomials u(n,x):
%e A208751   1
%e A208751   1 +  2x
%e A208751   1 +  6x +  2x^2
%e A208751   1 + 12x + 12x^2 +  2x^3
%e A208751   1 + 20x + 40x^2 + 18x^3 + 2x^4
%e A208751 From _Philippe Deléham_, Mar 17 2012: (Start)
%e A208751 (1, 0, 1, 0, 0, ...) DELTA (0, 2, -1, 0, 0, ...) begins:
%e A208751   1;
%e A208751   1,   0;
%e A208751   1,   2,   0;
%e A208751   1,   6,   2,   0;
%e A208751   1,  12,  12,   2,   0;
%e A208751   1,  20,  40,  18,   2,   0;
%e A208751   1,  30, 100,  86,  24,   2,   0; (End)
%t A208751 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208751 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208751 v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x];
%t A208751 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208751 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208751 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208751 TableForm[cu]
%t A208751 Flatten[%]    (* A208751 *)
%t A208751 Table[Expand[v[n, x]], {n, 1, z}]
%t A208751 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208751 TableForm[cv]
%t A208751 Flatten[%]    (* A208752 *)
%Y A208751 Cf. A208752, A208510.
%K A208751 nonn,tabl
%O A208751 1,3
%A A208751 _Clark Kimberling_, Mar 01 2012