cp's OEIS Frontend

A209696 Triangle of coefficients of polynomials v(n,x) jointly generated with A209695; see the Formula section.

%I A209696 #15 Jan 22 2020 20:13:45
%S A209696 1,1,3,1,6,7,1,9,23,17,1,12,48,76,41,1,15,82,204,233,99,1,18,125,428,
%T A209696 765,682,239,1,21,177,775,1907,2649,1935,577,1,24,238,1272,4010,7656,
%U A209696 8680,5368,1393,1,27,308,1946,7506,18358,28548,27312,14641,3363
%N A209696 Triangle of coefficients of polynomials v(n,x) jointly generated with A209695; see the Formula section.
%C A209696 Alternating row sums: 1,-2,2,-2,2,-2,2,-2,...
%C A209696 For a discussion and guide to related arrays, see A208510.
%C A209696 Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 24 2012
%C A209696 Mirror image of triangle in A054458. - _Philippe Deléham_, Mar 24 2012
%F A209696 u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
%F A209696 v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),
%F A209696 where u(1,x)=1, v(1,x)=1.
%F A209696 From _Philippe Deléham_, Mar 24 2012: (Start)
%F A209696 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209696 G.f.: (1-2*y*x-y^2*x^2)/(1-x-2*y*x-y*x^2-y^2*x^2).
%F A209696 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A209696 First five rows:
%e A209696   1;
%e A209696   1,  3;
%e A209696   1,  6,  7;
%e A209696   1,  9, 23, 17;
%e A209696   1, 12, 48, 76, 41;
%e A209696 First three polynomials v(n,x):
%e A209696   1
%e A209696   1 + 3x
%e A209696   1 + 6x + 7x^2.
%e A209696 From _Philippe Deléham_, Mar 24 2012: (Start)
%e A209696 (1, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, ...) begins:
%e A209696   1;
%e A209696   1,  0;
%e A209696   1,  3,  0;
%e A209696   1,  6,  7,  0;
%e A209696   1,  9, 23, 17,  0;
%e A209696   1, 12, 48, 76, 41,  0; (End)
%t A209696 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209696 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209696 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209696 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209696 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209696 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209696 TableForm[cu]
%t A209696 Flatten[%]    (* A209695 *)
%t A209696 Table[Expand[v[n, x]], {n, 1, z}]
%t A209696 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209696 TableForm[cv]
%t A209696 Flatten[%]    (* A209696 *)
%Y A209696 Cf. A054458, A209695, A208510.
%K A209696 nonn,tabl
%O A209696 1,3
%A A209696 _Clark Kimberling_, Mar 13 2012