|
%I #13 Jan 22 2020 20:13:06
%S 1,2,1,3,3,1,4,7,5,1,5,14,15,6,1,6,25,36,23,8,1,7,41,76,69,36,9,1,8,
%T 63,147,176,123,48,11,1,9,92,266,400,355,192,66,12,1,10,129,456,834,
%U 910,635,292,82,14,1,11,175,747,1626,2131,1833,1065,410,105,15,1
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208834; see the Formula section.
%C row sums, u(n,1): A000129
%C row sums, v(n,1): A001333
%C alternating row sums, u(n,-1): 1,0,-1,-2,-3,-4,-5,-6,...
%C alternating row sums, v(n,-1): 1,1,1,1,1,1,1,1,1,1,1,...
%C Subtriangle of the triangle T(n,k) given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 26 2012
%F u(n,x) = u(n-1,x) + x*v(n-1,x),
%F v(n,x) = (x+1)*u(n-1,x) + v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 26 2012: (Start)
%F As DELTA-triangle T(n,k) with 0 <= k <= n:
%F G.f.: (1-x+x^2-y^2*x^2)/(1-2*x+x^2-y*x^2-y^2*x^2).
%F T(n,k) = 2*T(n-1,k) - T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0. (End)
%e First five rows:
%e 1;
%e 2, 1;
%e 3, 3, 1;
%e 4, 7, 5, 1;
%e 5, 14, 15, 6, 1;
%e First five polynomials v(n,x):
%e 1
%e 2 + x
%e 3 + 3x + x^2
%e 4 + 7x + 5x^2 + x^3
%e 5 + 14x + 15x^2 + 6x^3 + x^4
%e From _Philippe Deléham_, Mar 26 2012: (Start)
%e (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 2, 1, 0;
%e 3, 3, 1, 0;
%e 4, 7, 5, 1, 0;
%e 5, 14, 15, 6, 1, 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] + 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[%] (* A208334 *)
%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[%] (* A208335 *)
%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)
%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)
%t Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)
%t Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)
%Y Cf. A208334.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Feb 26 2012
|
