|
%I #14 Jan 24 2020 03:27:17
%S 1,1,1,1,2,3,1,3,7,7,1,4,12,20,17,1,5,18,40,57,41,1,6,25,68,129,158,
%T 99,1,7,33,105,243,399,431,239,1,8,42,152,410,824,1200,1160,577,1,9,
%U 52,210,642,1506,2692,3528,3089,1393,1,10,63,280,952,2532,5290
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section.
%C Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 09 2012
%F u(n,x) = u(n-1,x) + x*v(n-1,x),
%F v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Apr 09 2012: (Start)
%F As DELTA-triangle T(n,k) with 0 <= k <= n:
%F G.f.: (1-2*y*x-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).
%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e First five rows:
%e 1;
%e 1, 1;
%e 1, 2, 3;
%e 1, 3, 7, 7;
%e 1, 4, 12, 20, 17;
%e First five polynomials u(n,x):
%e 1
%e 1 + x
%e 1 + 2x + 3x^2
%e 1 + 3x + 7x^2 + 7x^3
%e 1 + 4x + 12x^2 + 20x^3 + 17x^4
%e From _Philippe Deléham_, Apr 09 2012: (Start)
%e (1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 2, 3, 0;
%e 1, 3, 7, 7, 0;
%e 1, 4, 12, 20, 17, 0;
%e 1, 5, 18, 40, 57, 41, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t v[n_, x_] := (x + 1)*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[%] (* A208338 *)
%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[%] (* A208339 *)
%Y Cf. A208339.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Feb 27 2012
|
