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A209126 Triangle of coefficients of polynomials u(n,x) jointly generated with A209127; see the Formula section.

%I A209126 #14 Jan 24 2020 03:27:35
%S A209126 1,2,1,2,3,2,2,5,7,3,2,7,14,13,5,2,9,23,32,25,8,2,11,34,62,71,46,13,2,
%T A209126 13,47,105,156,149,84,21,2,15,62,163,295,367,304,151,34,2,17,79,238,
%U A209126 505,767,827,604,269,55,2,19,98,332,805,1435,1889,1798,1177,475
%N A209126 Triangle of coefficients of polynomials u(n,x) jointly generated with A209127; see the Formula section.
%C A209126 u(n,n) = A000045(n), Fibonacci numbers.
%C A209126 Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,...
%C A209126 For a discussion and guide to related arrays, see A208510.
%C A209126 Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 21 2012
%F A209126 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209126 v(n,x) = x*u(n-1,x) + x*v(n-1,x),
%F A209126 where u(1,x)=1, v(1,x)=1.
%F A209126 From _Philippe Deléham_, Mar 21 2012: (Start)
%F A209126 As DELTA-triangle with 0 <= k <= n:
%F A209126 G.f.: (1-y*x+x^2-y^2*x^2)/(1-x-y*x-y^2*x^2).
%F A209126 T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A209126 First five rows:
%e A209126   1;
%e A209126   2,  1;
%e A209126   2,  3,  2;
%e A209126   2,  5,  7,  3;
%e A209126   2,  7, 14, 13,  5;
%e A209126 First three polynomials u(n,x):
%e A209126   1
%e A209126   2 + x
%e A209126   2 + 3x + 2x^2
%e A209126 From _Philippe Deléham_, Mar 21 2012: (Start)
%e A209126 (1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
%e A209126   1;
%e A209126   1,  0;
%e A209126   2,  1,  0;
%e A209126   2,  3,  2,  0;
%e A209126   2,  5,  7,  3,  0;
%e A209126   2,  7, 14, 13,  5,  0; (End)
%t A209126 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209126 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209126 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
%t A209126 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209126 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209126 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209126 TableForm[cu]
%t A209126 Flatten[%]    (* A209126 *)
%t A209126 Table[Expand[v[n, x]], {n, 1, z}]
%t A209126 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209126 TableForm[cv]
%t A209126 Flatten[%]    (* A209127 *)
%Y A209126 Cf. A209127, A208510.
%K A209126 nonn,tabl
%O A209126 1,2
%A A209126 _Clark Kimberling_, Mar 05 2012