|
%I #16 Aug 11 2015 13:54:40
%S 1,1,2,1,2,8,1,2,12,24,1,2,16,40,80,1,2,20,56,160,256,1,2,24,72,256,
%T 576,832,1,2,28,88,368,992,2112,2688,1,2,32,104,496,1504,3968,7552,
%U 8704,1,2,36,120,640,2112,6464,15232,26880,28160,1,2,40,136,800
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208748; see the Formula section.
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle T(n,k) given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 14 2012
%F u(n,x)=u(n-1,x)+2x*v(n-1,x),
%F v(n,x)=2x*u(n-1,x)+2x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F T(n,k) = A208342(n,k)*2^k. - _Philippe Deléham_, Mar 05 2012
%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 14 2012
%F G.f.: -x*y/(-1+2*x*y-2*x^2*y+4*x^2*y^2+x). - _R. J. Mathar_, Aug 11 2015
%e First five rows:
%e 1
%e 1...2
%e 1...2...8
%e 1...2...12...24
%e 1...2...16...40...80
%e First five polynomials u(n,x):
%e 1
%e 1 + 2x
%e 1 + 2x + 8x^2
%e 1 + 2x + 12x^2 + 24x^3
%e 1 + 2x + 16x^2 + 40x^3 + 80x^4
%e (1, 0, -1, 1, 0, 0, ...) DELTA (0, 0, 0, -2, 0, 0, ...) begins :
%e 1
%e 1, 0
%e 1, 2, 0
%e 1, 2, 8, 0
%e 1, 2, 12, 24, 0
%e 1, 2, 16, 40, 80, 0
%e 1, 2, 20, 56, 160, 256, 0
%e 1, 2, 24, 72, 256, 576, 832, 0. - _Philippe Deléham_, Mar 14 2012
%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_] := 2 x*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[%] (* A208747 *)
%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[%] (* A208748 *)
%Y Cf. A208748, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 02 2012
|
