|
%I #23 Mar 31 2018 18:01:19
%S 1,1,2,1,4,6,1,6,18,14,1,8,36,56,38,1,10,60,140,190,94,1,12,90,280,
%T 570,564,246,1,14,126,490,1330,1974,1722,622,1,16,168,784,2660,5264,
%U 6888,4976,1606,1,18,216,1176,4788,11844,20664,22392,14454,4094,1
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208766; see the Formula section.
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 18 2012
%H G. C. Greubel, <a href="/A208765/b208765.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F u(n,x) = u(n-1,x) + 2*x*v(n-1,x),
%F v(n,x) = 2*x*u(n-1,x) + (x+1)*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 18 2012: (Start)
%F As DELTA-triangle with 0 <= k <= n:
%F G.f.: (1-x-y*x+2*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*y^2*x^2).
%F 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) = 2 and T(n,k) = 0 if k<0 or if k>n.
%F T(n,k) = binomial(n-1,k)*A026597(k). (End)
%e First five rows:
%e 1;
%e 1, 2;
%e 1, 4, 6;
%e 1, 6, 18, 14;
%e 1, 8, 36, 56, 38;
%e First five polynomials u(n,x):
%e 1
%e 1 + 2x
%e 1 + 4x + 6x^2
%e 1 + 6x + 18x^2 + 14x^3
%e 1 + 8x + 36x^2 + 56x^3 + 38x^4
%e (1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 4, 6, 0;
%e 1, 6, 18, 14, 0;
%e 1, 8, 36, 56, 38, 0;
%e 1, 10, 60, 140, 190, 94, 0. - _Philippe Deléham_, Mar 18 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] + (x + 1) 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[%] (* A208765 *)
%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[%] (* A208766 *)
%t Rest[CoefficientList[CoefficientList[Series[(1-x-y*x+2*y*x^2-4*y^2*x^2)/( 1-2*x-y*x+x^2+y*x^2-4*y^2*x^2), {x,0,20}, {y,0,20}], x], y]//Flatten] (* _G. C. Greubel_, Mar 28 2018 *)
%Y Cf. A208766, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 02 2012
|
