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A208749 Triangle of coefficients of polynomials u(n,x) jointly generated with A208750; see the Formula section.

%I A208749 #14 Jan 24 2020 03:30:30
%S A208749 1,1,2,1,6,2,1,12,10,4,1,20,32,24,4,1,30,80,88,36,8,1,42,170,256,180,
%T A208749 72,8,1,56,322,644,660,384,104,16,1,72,560,1456,1992,1568,704,192,16,
%U A208749 1,90,912,3024,5256,5360,3392,1344,272,32,1,110,1410,5856,12552
%N A208749 Triangle of coefficients of polynomials u(n,x) jointly generated with A208750; see the Formula section.
%C A208749 For a discussion and guide to related arrays, see A208510.
%C A208749 Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 14 2012
%F A208749 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208749 v(n,x) = (x+1)*u(n-1,x) + v(n-1,x),
%F A208749 where u(1,x)=1, v(1,x)=1.
%F A208749 As DELTA-triangle: g.f.: (1-x-2*y^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - _Philippe Deléham_, Mar 14 2012
%F A208749 Recurrence: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > =n. - _Philippe Deléham_, Mar 14 2012
%e A208749 First five rows:
%e A208749   1;
%e A208749   1,  2;
%e A208749   1,  6,  2;
%e A208749   1, 12, 10,  4;
%e A208749   1, 20, 32, 24,  4;
%e A208749 First five polynomials u(n,x):
%e A208749   1
%e A208749   1 +  2x
%e A208749   1 +  6x +  2x^2
%e A208749   1 + 12x + 10x^2 +  4x^3
%e A208749   1 + 20x + 32x^2 + 24x^3 + 4x^4
%e A208749 From _Philippe Deléham_, Mar 14 2012: (Start)
%e A208749 (1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, ...) begins:
%e A208749   1;
%e A208749   1,  0;
%e A208749   1,  2,  0;
%e A208749   1,  6,  2,  0;
%e A208749   1, 12, 10,  4,  0;
%e A208749   1, 20, 32, 24,  4,  0;
%e A208749   1, 30, 80, 88, 36,  8,  0; (End)
%t A208749 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208749 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208749 v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x];
%t A208749 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208749 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208749 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208749 TableForm[cu]
%t A208749 Flatten[%]    (* A208749 *)
%t A208749 Table[Expand[v[n, x]], {n, 1, z}]
%t A208749 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208749 TableForm[cv]
%t A208749 Flatten[%]    (* A208750 *)
%Y A208749 Cf. A208750, A208510.
%K A208749 nonn,tabl
%O A208749 1,3
%A A208749 _Clark Kimberling_, Mar 02 2012