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A209695 Triangle of coefficients of polynomials u(n,x) jointly generated with A209696; see the Formula section.

%I A209695 #19 Feb 17 2025 09:28:51
%S A209695 1,1,2,1,5,5,1,8,18,12,1,11,40,58,29,1,14,71,164,175,70,1,17,111,357,
%T A209695 601,507,169,1,20,160,664,1550,2048,1428,408,1,23,218,1112,3346,6106,
%U A209695 6632,3940,985,1,26,285,1728,6394,15012,22442,20680,10701,2378
%N A209695 Triangle of coefficients of polynomials u(n,x) jointly generated with A209696; see the Formula section.
%C A209695 Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
%C A209695 For a discussion and guide to related arrays, see A208510.
%C A209695 Subtriangle of the triangle given by (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 24 2012
%F A209695 u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
%F A209695 v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),
%F A209695 where u(1,x)=1, v(1,x)=1.
%F A209695 From _Philippe Deléham_, Mar 24 2012: (Start)
%F A209695 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209695 G.f.: (1-2*y*x-y*x^2-y^2*x^2)/(1-x-2*y*x-y*x^2-y^2*x^2).
%F A209695 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) + 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 A209695 First five rows:
%e A209695   1;
%e A209695   1,  2;
%e A209695   1,  5,  5;
%e A209695   1,  8, 18, 12;
%e A209695   1, 11, 40, 58, 29;
%e A209695 First three polynomials u(n,x):
%e A209695   1
%e A209695   1 + 2x
%e A209695   1 + 5x + 5x^2.
%e A209695 From _Philippe Deléham_, Mar 24 2012: (Start)
%e A209695 (1, 0, 1/2, -1/2, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, ...) begins:
%e A209695   1;
%e A209695   1,  0;
%e A209695   1,  2,  0;
%e A209695   1,  5,  5,  0;
%e A209695   1,  8, 18, 12,  0;
%e A209695   1, 11, 40, 58, 29, 0; (End)
%t A209695 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209695 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209695 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209695 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209695 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209695 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209695 TableForm[cu]
%t A209695 Flatten[%]    (* A209695 *)
%t A209695 Table[Expand[v[n, x]], {n, 1, z}]
%t A209695 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209695 TableForm[cv]
%t A209695 Flatten[%]    (* A209696 *)
%Y A209695 Cf. A209696, A208510, A000129, A008288.
%K A209695 nonn,tabl
%O A209695 1,3
%A A209695 _Clark Kimberling_, Mar 13 2012