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A208766 Triangle of coefficients of polynomials v(n,x) jointly generated with A208765; see the Formula section.

%I A208766 #18 Jan 24 2020 03:27:43
%S A208766 1,1,3,1,6,7,1,9,21,19,1,12,42,76,47,1,15,70,190,235,123,1,18,105,380,
%T A208766 705,738,311,1,21,147,665,1645,2583,2177,803,1,24,196,1064,3290,6888,
%U A208766 8708,6424,2047,1,27,252,1596,5922,15498,26124,28908,18423
%N A208766 Triangle of coefficients of polynomials v(n,x) jointly generated with A208765; see the Formula section.
%C A208766 For a discussion and guide to related arrays, see A208510.
%C A208766 Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -4/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 20 2012
%F A208766 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208766 v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),
%F A208766 where u(1,x)=1, v(1,x)=1.
%F A208766 From _Philippe Deléham_, Mar 20 2012: (Start)
%F A208766 As DELTA-triangle with 0 <= k <= n:
%F A208766 G.f.: (1-x-y*x+3*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*y^2*x^2).
%F A208766 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 4*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 A208766 First five rows:
%e A208766   1;
%e A208766   1,  3;
%e A208766   1,  6,  7;
%e A208766   1,  9, 21, 19;
%e A208766   1, 12, 42, 76, 47;
%e A208766 First five polynomials v(n,x):
%e A208766   1
%e A208766   1 +  3x
%e A208766   1 +  6x +  7x^2
%e A208766   1 +  9x + 21x^2 + 19x^3
%e A208766   1 + 12x + 42x^2 + 76x^3 + 47x^4
%e A208766 From _Philippe Deléham_, Mar 20 2012: (Start)
%e A208766 (1, 0, 0, 1, 0, 0, ...) DELTA (0, 3, -2/3, -4/3, 0, 0, ...) begins:
%e A208766   1;
%e A208766   1,  0;
%e A208766   1,  3,  0;
%e A208766   1,  6,  7,  0;
%e A208766   1,  9, 21, 19,  0;
%e A208766   1, 12, 42, 76, 47,  0; (End)
%t A208766 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208766 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208766 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1) v[n - 1, x];
%t A208766 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208766 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208766 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208766 TableForm[cu]
%t A208766 Flatten[%]     (* A208765 *)
%t A208766 Table[Expand[v[n, x]], {n, 1, z}]
%t A208766 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208766 TableForm[cv]
%t A208766 Flatten[%]     (* A208766 *)
%Y A208766 Cf. A208765, A208510.
%K A208766 nonn,tabl
%O A208766 1,3
%A A208766 _Clark Kimberling_, Mar 02 2012