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A209171 Triangle of coefficients of polynomials v(n,x) jointly generated with A209170; see the Formula section.

%I A209171 #19 Jan 24 2020 03:33:34
%S A209171 1,3,2,6,8,3,12,25,19,5,24,68,77,40,8,48,172,259,201,80,13,96,416,782,
%T A209171 806,478,154,21,192,976,2200,2825,2222,1067,289,34,384,2240,5888,9048,
%U A209171 8857,5640,2277,532,55,768,5056,15184,27160,31787,25184,13483
%N A209171 Triangle of coefficients of polynomials v(n,x) jointly generated with A209170; see the Formula section.
%C A209171 Column 1:  Fibonacci numbers (A000045).
%C A209171 Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,...
%C A209171 For a discussion and guide to related arrays, see A208510.
%C A209171 Subtriangle of (1, 2, -3/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 10 2012
%F A209171 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209171 v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x)+1,
%F A209171 where u(1,x)=1, v(1,x)=1.
%F A209171 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 3, T(2,1) = 2. - _Philippe Deléham_, Mar 10 2012
%F A209171 Sum_{k=0..n} T(n,k)*x^k = A000012(n), A003945(n-1), A007483(n-1) for x = -1, 0, 1 respectively. - _Philippe Deléham_, Mar 10 2012
%F A209171 G.f.: (-1-x-x*y)*x*y/(-1+2*x+x*y+x^2*y^2+x^2*y). - _R. J. Mathar_, Aug 12 2015
%e A209171 First five rows:
%e A209171    1;
%e A209171    3,  2;
%e A209171    6,  8,  3;
%e A209171   12, 25, 19,  5;
%e A209171   24, 68, 77, 40,  8;
%e A209171 First three polynomials v(n,x):
%e A209171   1
%e A209171   3 + 2x
%e A209171   6 + 8x + 3x^2.
%e A209171 From _Philippe Deléham_, Mar 10 2012: (Start)
%e A209171 Triangle (1, 2, -3/2, 1/2, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins (0 <= k <= n):
%e A209171    1;
%e A209171    1,   0;
%e A209171    3,   2,   0;
%e A209171    6,   8,   3,   0;
%e A209171   12,  25,  19,   5,   0;
%e A209171   24,  68,  77,  40,   8,   0;
%e A209171   48, 172, 259, 201,  80,  13,   0;
%e A209171   96, 416, 782, 806, 478, 154,  21,   0; (End)
%t A209171 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209171 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209171 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209171 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209171 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209171 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209171 TableForm[cu]
%t A209171 Flatten[%]    (* A209170 *)
%t A209171 Table[Expand[v[n, x]], {n, 1, z}]
%t A209171 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209171 TableForm[cv]
%t A209171 Flatten[%]    (* A209171 *)
%Y A209171 Cf. A209170, A208510.
%K A209171 nonn,tabl
%O A209171 1,2
%A A209171 _Clark Kimberling_, Mar 08 2012