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

%I A209140 #18 Jan 27 2020 00:43:32
%S A209140 1,1,3,2,5,7,3,12,18,17,5,23,51,58,41,8,45,118,189,175,99,13,84,264,
%T A209140 506,645,507,239,21,155,558,1268,1950,2085,1428,577,34,281,1145,2974,
%U A209140 5395,6998,6482,3940,1393,55,504,2286,6687,13851,21141,23856
%N A209140 Triangle of coefficients of polynomials v(n,x) jointly generated with A209139; see the Formula section.
%C A209140 Column 1:  Fibonacci numbers, A000045.
%C A209140 Alternating row sums: (-2)^(n-1).
%C A209140 For a discussion and guide to related arrays, see A208510.
%F A209140 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209140 v(n,x) = (x+1)*u(n-1,x) + x*v(n-1,x),
%F A209140 where u(1,x)=1, v(1,x)=1.
%F A209140 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 3, T(n,k) = 0 if k < 0 or if k >= n. - _Philippe Deléham_, Apr 11 2012
%e A209140 First five rows:
%e A209140   1;
%e A209140   1,  3;
%e A209140   2,  5,  7;
%e A209140   3, 12, 18, 17;
%e A209140   5, 23, 51, 58, 41;
%e A209140 First three polynomials v(n,x):
%e A209140   1
%e A209140   1 + 3x
%e A209140   2 + 5x + 7x^2.
%t A209140 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209140 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209140 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A209140 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209140 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209140 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209140 TableForm[cu]
%t A209140 Flatten[%]    (* A209139 *)
%t A209140 Table[Expand[v[n, x]], {n, 1, z}]
%t A209140 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209140 TableForm[cv]
%t A209140 Flatten[%]    (* A209140 *)
%Y A209140 Cf. A209139, A208510.
%K A209140 nonn,tabl
%O A209140 1,3
%A A209140 _Clark Kimberling_, Mar 05 2012