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%I #35 Apr 13 2020 06:23:54
%S 1,2,3,2,5,4,4,8,10,8,8,13,20,24,16,16,21,40,52,56,32,32,34,76,116,
%T 128,128,64,64,55,142,240,312,304,288,128,128,89,260,488,688,800,704,
%U 640,256,256,144,470,964,1496,1856,1984,1600,1408,512,512,233,840
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210596; see the Formula section.
%C Row sums: even-indexed Fibonacci numbers.
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 25 2012
%H G. C. Greubel, <a href="/A210221/b210221.txt">Rows n = 1..100 of triangle, flattened</a>
%F u(n,x) = u(n-1,x) + v(n-1,x),
%F v(n,x) = u(n-1,x) + 2*x*v(n-1,x) [Corrected by _Indranil Ghosh_, May 27 2017]
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 25 2012: (Start)
%F As DELTA-triangle T(n,k) with 0 <= k <= n:
%F G.f.: (1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2).
%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 2, T(1,1) = T(2,1) = T(2,2) = 0, T(n,k) = 0 if k < 0 or if k >= n. (End)
%e First five rows:
%e 1;
%e 2;
%e 3, 2;
%e 5, 4, 4;
%e 8, 10, 8, 8;
%e First three polynomials u(n,x):
%e 1
%e 2
%e 3 + 2x.
%e From _Philippe Deléham_, Mar 25 2012: (Start)
%e (1, 1, -1, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 2, 0, 0;
%e 3, 2, 0, 0;
%e 5, 4, 4, 0, 0;
%e 8, 10, 8, 8, 0, 0;
%e 13, 20, 24, 16, 16, 0, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, 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[%] (* A210221 *)
%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[%] (* A210596 *)
%t With[{m = 10}, Rest[CoefficientList[CoefficientList[Series[(1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2), {x, 0, m}, {y, 0, m}], x], y]]]//Flatten (* _G. C. Greubel_, Dec 16 2018 *)
%t T[n_, k_]:= If[k < 0 || k > n, 0, T[n-1, k] + 2*T[n-1, k-1] + T[n-2, k] - 2*T[n-2, k-1]]; T[1, 0] = 1 ; T[2, 0] = 2; T[2, 1] = 0; Join[{1}, Table[T[n, k], {n, 1, 10}, {k, 0, n-2}]//Flatten] (* _G. C. Greubel_, Dec 17 2018 *)
%o (Python)
%o from sympy import Poly
%o from sympy.abc import x
%o def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
%o def v(n, x): return 1 if n==1 else u(n - 1, x) + 2*x*v(n - 1, x)
%o def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
%o for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 27 2017
%Y Cf. A210596, A208510.
%K nonn,tabf
%O 1,2
%A _Clark Kimberling_, Mar 24 2012
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