cp's OEIS Frontend

A209138 Triangle of coefficients of polynomials v(n,x) jointly generated with A209137; see the Formula section.

%I A209138 #18 Jan 22 2020 03:35:41
%S A209138 1,1,2,2,4,3,3,9,10,5,5,18,28,22,8,8,35,68,74,45,13,13,66,154,210,177,
%T A209138 88,21,21,122,331,541,574,397,167,34,34,222,686,1302,1656,1446,850,
%U A209138 310,55,55,399,1382,2982,4404,4614,3434,1758,566,89,89,710,2723
%N A209138 Triangle of coefficients of polynomials v(n,x) jointly generated with A209137; see the Formula section.
%C A209138 Every row begins and ends with a Fibonacci number (A000045).
%C A209138 u(n,1) = n-th row sum = 3^(n-1).
%C A209138 Alternating row sums: 1,-1,1,-1,1,-1,1,-1,1,-1,...
%C A209138 For a discussion and guide to related arrays, see A208510.
%F A209138 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209138 v(n,x) = (x+1)*u(n-1,x) + x*v(n-1,x),
%F A209138 where u(1,x)=1, v(1,x)=1.
%F A209138 From _Philippe Deléham_, Apr 11 2012: (Start)
%F A209138 T(n,k) = A185081(n,k+1).
%F A209138 T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k >= n. (End)
%e A209138 First five rows:
%e A209138   1;
%e A209138   1,  2;
%e A209138   2,  4,  3;
%e A209138   3,  9, 10,  5;
%e A209138   5, 18, 28, 22,  8;
%e A209138 First three polynomials v(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.
%e A209138 From _Philippe Deléham_, Apr 11 2012: (Start)
%e A209138 Triangle in A185081 begins:
%e A209138   1;
%e A209138   0,  1;
%e A209138   0,  1,  2;
%e A209138   0,  2,  4,  3;
%e A209138   0,  3,  9, 10,  5;
%e A209138   0,  5, 18, 28, 22,  8;
%e A209138   ... (End)
%t A209138 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209138 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209138 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x];
%t A209138 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209138 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209138 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209138 TableForm[cu]
%t A209138 Flatten[%]    (* A209137 *)
%t A209138 Table[Expand[v[n, x]], {n, 1, z}]
%t A209138 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209138 TableForm[cv]
%t A209138 Flatten[%]    (* A209138 *)
%Y A209138 Cf. A209137, A208510.
%K A209138 nonn,tabl
%O A209138 1,3
%A A209138 _Clark Kimberling_, Mar 05 2012