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A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.

%I A208343 #20 Jan 26 2020 21:07:45
%S A208343 1,0,2,0,1,3,0,1,2,5,0,1,2,5,8,0,1,2,6,10,13,0,1,2,7,13,20,21,0,1,2,8,
%T A208343 16,29,38,34,0,1,2,9,19,39,60,71,55,0,1,2,10,22,50,86,122,130,89,0,1,
%U A208343 2,11,25,62,116,187,241,235,144,0,1,2,12,28,75,150,267,392,468
%N A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.
%C A208343 u(n,n) = A000045(n+1) (Fibonacci numbers).
%C A208343 n-th row sum:  2^(n-1)
%C A208343 As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 26 2012
%F A208343 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A208343 v(n,x) = x*u(n-1,x) + x*v(n-1,x),
%F A208343 where u(1,x)=1, v(1,x)=1.
%F A208343 From _Philippe Deléham_, Feb 26 2012: (Start)
%F A208343 As triangle T(n,k) with 0 <= k <= n:
%F A208343 T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k > n or if k < 0.
%F A208343 G.f.: (1-(1-y)*x)/(1-(1+y)*x+y*(1-y)*x^2).
%F A208343 Sum_{k=0..n} T(n,k)*x^k = (-1)*A091003(n+1), A152166(n), A000007(n), A000079(n), A055099(n), A152224(n) for x = -2, -1, 0, 1, 2, 3 respectively.
%F A208343 Sum_{k=0..n} T(n,k)*x^(n-k) = A087205(n), A140165(n+1), A016116(n+1), A000045(n+2), A000079(n), A122367(n), A006012(n), A052961(n), A154626(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively. (End)
%F A208343 T(n,k) = A208748(n,k)/2^k. - _Philippe Deléham_, Mar 05 2012
%e A208343 First five rows:
%e A208343   1;
%e A208343   0, 2;
%e A208343   0, 1, 3;
%e A208343   0, 1, 2, 5;
%e A208343   0, 1, 2, 5, 8;
%e A208343 First five polynomials v(n,x):
%e A208343   1
%e A208343      2x
%e A208343       x + 3x^2
%e A208343       x + 2x^2 + 5x^3
%e A208343       x + 2x^2 + 5x^3 + 8x^4.
%t A208343 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208343 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208343 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
%t A208343 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208343 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208343 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208343 TableForm[cu]
%t A208343 Flatten[%]  (* A208342 *)
%t A208343 Table[Expand[v[n, x]], {n, 1, z}]
%t A208343 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208343 TableForm[cv]
%t A208343 Flatten[%]  (* A208343 *)
%Y A208343 Cf. A208343.
%Y A208343 Cf. A084938, A000045, A000079.
%K A208343 nonn,tabl
%O A208343 1,3
%A A208343 _Clark Kimberling_, Feb 25 2012