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%I #26 Jan 24 2020 03:29:43
%S 1,1,2,1,3,4,1,5,7,8,1,6,17,15,16,1,8,23,49,31,32,1,9,39,72,129,63,64,
%T 1,11,48,150,201,321,127,128,1,12,70,198,501,522,769,255,256,1,14,82,
%U 338,699,1524,1291,1793,511,512,1,15,110,420,1375,2223,4339,3084,4097,1023,1024
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209172; see the Formula section.
%C For n > 1, n-th alternating row sum = ((-1)^(n-1)*F(2n-3), where F=A000045 (Fibonacci numbers).
%C Coefficient of x^(n-1) in u(n,x): 2^(n-1).
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 11 2012
%H G. C. Greubel, <a href="/A209413/b209413.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F u(n,x) = x*u(n-1,x) + v(n-1,x),
%F v(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 11 2012: (Start)
%F As DELTA-triangle T(n,k) with 0 <= k <= n:
%F T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(2,0) = 1, T(2,1) = 2, T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.
%F G.f.: (1+x-3*y*x-y*x^2+2*y^2*x^2)/(1-3*y*x-(1-2y^2)*x^2). (End)
%e First five rows:
%e 1;
%e 1, 2;
%e 1, 3, 4;
%e 1, 5, 7, 8;
%e 1, 6, 17, 15, 16;
%e First three polynomials v(n,x):
%e 1
%e 1 + 2x
%e 1 + 3x + 4x^2.
%e From _Philippe Deléham_, Mar 11 2012: (Start)
%e (1, 0, -1/2, -1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 3, 4, 0;
%e 1, 5, 7, 8, 0;
%e 1, 6, 17, 15, 16, 0;
%e 1, 8, 23, 49, 31, 32, 0;
%e 1, 9, 39, 72, 129, 63, 64, 0;
%e 1, 11, 48, 150, 201, 321, 127, 128, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A209172 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A209413 *)
%t CoefficientList[CoefficientList[Series[(1 + x - 3*y*x - y*x^2 + 2*y^2*x^2)/(1 - 3*y*x - (1 - 2 y^2)*x^2), {x,0,10}, {y,0,10}], x],y] // Flatten (* _G. C. Greubel_, Jan 03 2018 *)
%Y Cf. A209172, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 08 2012
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