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A209172 Triangle of coefficients of polynomials u(n,x) jointly generated with A209413; see the Formula section.

%I A209172 #15 Jan 24 2020 03:29:52
%S A209172 1,1,1,1,3,1,1,4,7,1,1,6,11,15,1,1,7,23,26,31,1,1,9,30,72,57,63,1,1,
%T A209172 10,48,102,201,120,127,1,1,12,58,198,303,522,247,255,1,1,13,82,256,
%U A209172 699,825,1291,502,511,1,1,15,95,420,955,2223,2116,3084,1013,1023,1
%N A209172 Triangle of coefficients of polynomials u(n,x) jointly generated with A209413; see the Formula section.
%C A209172 For n > 1, n-th alternating row sum = ((-1)^n)*F(2n-4), where F=A000045 (Fibonacci numbers).  For a discussion and guide to related arrays, see A208510.
%C A209172 Subtriangle of the triangle given by (1, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 11 2012
%F A209172 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209172 v(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A209172 where u(1,x)=1, v(1,x)=1.
%F A209172 From _Philippe Deléham_, Mar 11 2012: (Start)
%F A209172 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209172 T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-2) with T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.
%F A209172 G.f.: (1+x-3*y*x-2*y*x^2+2*y^2*x^2)/(1-3*y*x-x^2+2*y^2*x^2). (End)
%e A209172 First five rows:
%e A209172   1;
%e A209172   1,  1;
%e A209172   1,  3,  1;
%e A209172   1,  4,  7,  1;
%e A209172   1,  6, 11, 15,  1;
%e A209172 First three polynomials v(n,x):
%e A209172   1
%e A209172   1 + x
%e A209172   1 + 3x + x^2.
%e A209172 From _Philippe Deléham_, Mar 11 2012: (Start)
%e A209172 (1, 0, 1, -2, 0, 0, 0,...) DELTA (0, 1, 0, 2, 0, 0, ...) begins:
%e A209172   1;
%e A209172   1,  0;
%e A209172   1,  1,  0;
%e A209172   1,  3,  1,  0;
%e A209172   1,  4,  7,  1,  0;
%e A209172   1,  6, 11, 15,  1,  0;
%e A209172   1,  7, 23, 26, 31,  1,  0;
%e A209172   1,  9, 30, 72, 57, 63,  1,  0; (End)
%t A209172 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209172 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209172 v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A209172 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209172 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209172 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209172 TableForm[cu]
%t A209172 Flatten[%]    (* A209172 *)
%t A209172 Table[Expand[v[n, x]], {n, 1, z}]
%t A209172 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209172 TableForm[cv]
%t A209172 Flatten[%]    (* A209413 *)
%Y A209172 Cf. A209413, A208510.
%K A209172 nonn,tabl
%O A209172 1,5
%A A209172 _Clark Kimberling_, Mar 08 2012