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A208759 Triangle of coefficients of polynomials u(n,x) jointly generated with A208760; see the Formula section.

%I A208759 #20 Jan 24 2020 03:28:13
%S A208759 1,1,2,1,4,6,1,6,16,16,1,8,30,56,44,1,10,48,128,188,120,1,12,70,240,
%T A208759 504,608,328,1,14,96,400,1080,1872,1920,896,1,16,126,616,2020,4512,
%U A208759 6672,5952,2448,1,18,160,896,3444,9352,17856,23040,18192,6688,1,20,198,1248,5488,17472,40600,67776,77616,54976,18272
%N A208759 Triangle of coefficients of polynomials u(n,x) jointly generated with A208760; see the Formula section.
%C A208759 For a discussion and guide to related arrays, see A208510.
%C A208759 Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 18 2012
%H A208759 G. C. Greubel, <a href="/A208759/b208759.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F A208759 u(n,x) = u(n-1,x) + 2*x*v(n-1,x),
%F A208759 v(n,x) = (x+1)*u(n-1,x) + 2*x*v(n-1,x),
%F A208759 where u(1,x)=1, v(1,x)=1.
%F A208759 From _Philippe Deléham_, Mar 18 2012: (Start)
%F A208759 As DELTA-triangle with 0 <= k <= n:
%F A208759 G.f.: (1-2y*x-2*y^2*x^2)/(1-x-2*y*x-2*y^2*x^2).
%F A208759 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) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A208759 First five rows:
%e A208759   1;
%e A208759   1,  2;
%e A208759   1,  4,  6;
%e A208759   1,  6, 16, 16;
%e A208759   1,  8, 30, 56, 44;
%e A208759 First five polynomials u(n,x):
%e A208759   1
%e A208759   1 + 2x
%e A208759   1 + 4x +  6x^2
%e A208759   1 + 6x + 16x^2 + 16x^3
%e A208759   1 + 8x + 30x^2 + 56x^3 + 44x^4
%e A208759 From _Philippe Deléham_, Mar 18 2012: (Start)
%e A208759 (1, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, ...) begins:
%e A208759   1;
%e A208759   1,   0;
%e A208759   1,   2,   0;
%e A208759   1,   4,   6,   0;
%e A208759   1,   6,  16,  16,   0;
%e A208759   1,   8,  30,  56,  44,   0;
%e A208759   1,  10,  48, 128, 188, 120,  0; (End)
%t A208759 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208759 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208759 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A208759 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208759 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208759 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208759 TableForm[cu]
%t A208759 Flatten[%]  (* A208759 *)
%t A208759 Table[Expand[v[n, x]], {n, 1, z}]
%t A208759 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208759 TableForm[cv]
%t A208759 Flatten[%]  (* A208760 *)
%t A208759 Rest[CoefficientList[CoefficientList[Series[(1-2*y*x-2*y^2*x^2)/(1-x-2*y*x- 2*y^2*x^2), {x,0,20}, {y,0,20}], x], y]//Flatten] (* _G. C. Greubel_, Mar 28 2018 *)
%Y A208759 Cf. A208760, A208510.
%K A208759 nonn,tabl
%O A208759 1,3
%A A208759 _Clark Kimberling_, Mar 02 2012
%E A208759 Terms a(58) onward added by _G. C. Greubel_, Mar 28 2018