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%I #14 Jan 24 2020 03:28:06
%S 1,1,3,1,5,8,1,7,20,22,1,9,36,72,60,1,11,56,158,244,164,1,13,80,288,
%T 632,796,448,1,15,108,470,1320,2376,2528,1224,1,17,140,712,2420,5592,
%U 8544,7872,3344,1,19,176,1022,4060,11372,22368,29712,24144,9136
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208759; see the Formula section.
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle given by (1, 0, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 18 2012
%F u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 18 2012: (Start)
%F As DELTA-triangle with 0 <= k <= n:
%F G.f.: (1-2*y*x+y*x^2-2*y^2*x^2)/(1-x-2*y*x-2*y^2*x^2).
%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e First five rows:
%e 1;
%e 1, 3;
%e 1, 5, 8;
%e 1, 7, 20, 22;
%e 1, 9, 36, 72, 60;
%e First five polynomials v(n,x):
%e 1
%e 1 + 3x
%e 1 + 5x + 8x^2
%e 1 + 7x + 20x^2 + 22x^3
%e 1 + 9x + 36x^2 + 72x^3 + 60x^4
%e From _Philippe Deléham_, Mar 18 2012: (Start)
%e (1, 0, -1/3, 1/3, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 3, 0;
%e 1, 5, 8, 0;
%e 1, 7, 20, 22, 0;
%e 1, 9, 36, 72, 60, 0;
%e 1, 11, 56, 158, 244, 164, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t v[n_, x_] := (x + 1)*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[%] (* A208759 *)
%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[%] (* A208760 *)
%Y Cf. A208759, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 02 2012
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