cp's OEIS Frontend

A209139 Triangle of coefficients of polynomials u(n,x) jointly generated with A209140; see the Formula section.

%I A209139 #13 Jan 24 2020 03:25:13
%S A209139 1,2,1,3,5,3,5,12,15,7,8,27,45,42,17,13,55,119,151,116,41,21,108,282,
%T A209139 458,480,315,99,34,205,630,1228,1631,1467,845,239,55,381,1343,3054,
%U A209139 4849,5502,4358,2244,577,89,696,2769,7173,13218,17895,17838,12666
%N A209139 Triangle of coefficients of polynomials u(n,x) jointly generated with A209140; see the Formula section.
%C A209139 Column 1:  A000045 (Fibonacci numbers).
%C A209139 Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1...
%C A209139 For a discussion and guide to related arrays, see A208510.
%C A209139 Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 11 2012
%F A209139 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209139 v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F A209139 where u(1,x)=1, v(1,x)=1.
%F A209139 From _Philippe Deléham_, Apr 11 2012: (Start)
%F A209139 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209139 G.f.: (1-2*y*x-y*x^2-y^2*x^2)/(1-x-x^2-2*y*x-y^2*x^2).
%F A209139 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A209139 First five rows:
%e A209139   1;
%e A209139   2,  1;
%e A209139   3,  5,  3;
%e A209139   5, 12, 15,  7;
%e A209139   8, 27, 45, 42, 17;
%e A209139 First three polynomials u(n,x):
%e A209139   1
%e A209139   2 + x
%e A209139   3 + 5x + 3x^2
%e A209139 From _Philippe Deléham_, Apr 11 2012: (Start)
%e A209139 (1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
%e A209139    1;
%e A209139    1,  0;
%e A209139    2,  1,   0;
%e A209139    3,  5,   3,   0;
%e A209139    5, 12,  15,   7,   0;
%e A209139    8, 27,  45,  42,  17,  0;
%e A209139   13, 55, 119, 151, 116, 41, 0; (End)
%t A209139 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209139 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209139 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A209139 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209139 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209139 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209139 TableForm[cu]
%t A209139 Flatten[%]    (* A209139 *)
%t A209139 Table[Expand[v[n, x]], {n, 1, z}]
%t A209139 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209139 TableForm[cv]
%t A209139 Flatten[%]    (* A209140 *)
%Y A209139 Cf. A209140, A208510.
%K A209139 nonn,tabl
%O A209139 1,2
%A A209139 _Clark Kimberling_, Mar 05 2012