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A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.

%I A208345 #16 Aug 15 2025 03:49:51
%S A208345 1,0,3,0,1,7,0,1,3,17,0,1,3,10,41,0,1,3,11,30,99,0,1,3,12,35,87,239,0,
%T A208345 1,3,13,40,108,245,577,0,1,3,14,45,130,322,676,1393,0,1,3,15,50,153,
%U A208345 406,938,1836,3363,0,1,3,16,55,177,497,1236,2682,4925,8119,0,1
%N A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.
%C A208345 Row sums, u(n,1): (1,2,5,13,...), odd-indexed Fibonacci numbers.
%C A208345 Row sums, v(n,1): (1,3,8,21,...), even-indexed Fibonacci numbers.
%C A208345 As triangle T(n,k) with 0<=k<=n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 26 2012
%F A208345 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A208345 v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
%F A208345 where u(1,x)=1, v(1,x)=1.
%F A208345 From _Philippe Deléham_, Feb 26 2012: (Start)
%F A208345 As triangle T(n,k), 0<=k<=n:
%F A208345 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-2) - 2*T(n-2,k-1) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n.
%F A208345 G.f.: (1+(y-1)*x)/(1-(1+2*y)*x+y*(2-y)*x^2).
%F A208345 Sum_{k=0..n} T(n,k)*x^k = A152167(n), A000007(n), A001906(n+1), A003948(n) for x = -1, 0, 1, 2 respectively.
%F A208345 Sum_{k=0..n} T(n,k)*x^(n-k) = A078057(n), A001906(n+1), A000244(n), A081567(n), A083878(n), A165310(n) for x = 0, 1, 2, 3, 4, 5 respectively. (End)
%e A208345 First five rows:
%e A208345   1
%e A208345   0   3
%e A208345   0   1   7
%e A208345   0   1   3   17
%e A208345   0   1   3   10   41
%e A208345 First five polynomials u(n,x):
%e A208345   1, 3*x, x + 7*x^2, x + 3*x^2 + 17*x^3, x + 3*x^2 + 10*x^3 + 41*x^4.
%t A208345 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208345 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208345 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
%t A208345 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208345 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208345 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208345 TableForm[cu]
%t A208345 Flatten[%]    (* A208344 *)
%t A208345 Table[Expand[v[n, x]], {n, 1, z}]
%t A208345 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208345 TableForm[cv]
%t A208345 Flatten[%]    (* A208345 *)
%t A208345 Table[u[n, x] /. x -> 1, {n, 1, z}]
%t A208345 Table[v[n, x] /. x -> 1, {n, 1, z}]
%Y A208345 Cf. A208344.
%Y A208345 Cf. A084938, A000045, A000244, A001906.
%K A208345 nonn,tabl
%O A208345 1,3
%A A208345 _Clark Kimberling_, Feb 25 2012