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A208761 Triangle of coefficients of polynomials u(n,x) jointly generated with A208762; see the Formula section.

%I A208761 #20 Jan 24 2020 03:27:59
%S A208761 1,1,2,1,6,4,1,12,18,8,1,20,52,50,16,1,30,120,186,126,32,1,42,240,534,
%T A208761 576,306,64,1,56,434,1302,1986,1654,718,128,1,72,728,2828,5712,6632,
%U A208761 4484,1650,256,1,90,1152,5628,14436,21912,20508,11682,3726,512
%N A208761 Triangle of coefficients of polynomials u(n,x) jointly generated with A208762; see the Formula section.
%C A208761 Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,...
%C A208761 For a discussion and guide to related arrays, see A208510.
%C A208761 Subtriangle of the triangle given by [1,0,1,0,0,0,0,...] DELTA [0,2,0,-1,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 04 2012
%F A208761 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208761 v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x),
%F A208761 where u(1,x)=1, v(1,x)=1.
%F A208761 Recurrence: T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2). - _Philippe Deléham_, Mar 04 2012
%F A208761 G.f.: (-1-x*y+x)*x*y/(-1+x*y+2*x+2*x^2*y^2+x^2*y-x^2). - _R. J. Mathar_, Aug 12 2015
%e A208761 First five rows:
%e A208761   1;
%e A208761   1,  2;
%e A208761   1,  6,  4;
%e A208761   1, 12, 18,  8;
%e A208761   1, 20, 52, 50, 16;
%e A208761 First five polynomials u(n,x):
%e A208761   1
%e A208761   1 +  2x
%e A208761   1 +  6x +  4x^2
%e A208761   1 + 12x + 18x^2 +  8x^3
%e A208761   1 + 20x + 52x^2 + 50x^3 + 16x^4
%e A208761 From _Philippe Deléham_, Mar 04 2012: (Start)
%e A208761 Triangle (1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, ...) begins:
%e A208761   1;
%e A208761   1,   0;
%e A208761   1,   2,   0;
%e A208761   1,   6,   4,   0;
%e A208761   1,  12,  18,   8,   0;
%e A208761   1,  20,  52,  50,  16,   0;
%e A208761   1,  30, 120, 186, 126,  32,  0; (End)
%t A208761 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208761 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208761 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1) v[n - 1, x];
%t A208761 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208761 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208761 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208761 TableForm[cu]
%t A208761 Flatten[%]    (* A208761 *)
%t A208761 Table[Expand[v[n, x]], {n, 1, z}]
%t A208761 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208761 TableForm[cv]
%t A208761 Flatten[%]    (* A208762 *)
%Y A208761 Cf. A208762, A208510.
%K A208761 nonn,tabl
%O A208761 1,3
%A A208761 _Clark Kimberling_, Mar 03 2012