|
%I #10 Sep 08 2013 19:59:30
%S 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 1,3,13,40,108,245,577,0,1,3,14,45,130,322,676,1393,0,1,3,15,50,153,
%U 406,938,1836,3363,0,1,3,16,55,177,497,1236,2682,4925,8119,0,1
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.
%C row sums, u(n,1): (1,2,5,13,...), odd-indexed Fibonacci numbers
%C row sums, v(n,1): (1,3,8,21,...), even-indexed Fibonacci numbers
%C 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 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F v(n,x)=x*u(n-1,x)+2x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F (Start)- As triangle T(n,k), 0<=k<=n :
%F 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 G.f.: (1+(y-1)*x)/(1-(1+2*y)*x+y*(2-y)*x^2).
%F Sum_{k, 0<=k<=n} T(n,k)*x^k = A152167(n), A000007(n), A001906(n+1), A003948(n) for x = -1, 0, 1, 2 respectively.
%F Sum_{k, 0<=k<=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) - _Philippe Deléham_, Feb 26 2012
%e First five rows:
%e 1
%e 0...3
%e 0...1...7
%e 0...1...3...17
%e 0...1...3...10...41
%e First five polynomials u(n,x):
%e 1, 3x, x + 7x^2, x + 3x^2 + 17x^3, x + 3x^2 + 10x^3 +
%e 41x^4.
%t u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A208344 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A208345 *)
%t Table[u[n, x] /. x -> 1, {n, 1, z}]
%t Table[v[n, x] /. x -> 1, {n, 1, z}]
%Y Cf. A208344.
%Y Cf. A084938, A000045, A000244, A001906
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Feb 25 2012
|
