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Triangle of coefficients of polynomials v(n,x) jointly generated with A208328; see the Formula section.
3

%I #19 Oct 26 2024 03:32:16

%S 1,0,3,0,2,5,0,2,4,11,0,2,4,14,21,0,2,4,18,32,43,0,2,4,22,44,82,85,0,

%T 2,4,26,56,130,188,171,0,2,4,30,68,186,324,438,341,0,2,4,34,80,250,

%U 492,834,984,683,0,2,4,38,92,322,692,1374,2028,2202,1365,0,2,4,42

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208328; see the Formula section.

%C Row sums, u(n,1): A000129

%C Row sums, v(n,1): A001333

%C As triangle T(n,k) with 0 <= k <= n, it is (0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 27 2012

%F u(n,x) = u(n-1,x) + x*v(n-1,x),

%F v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),

%F where u(1,x)=1, v(1,x)=1.

%F From _Philippe Deléham_, Feb 27 2012: (Start)

%F As triangle T(n,k), 0 <= k <= n:

%F T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3 and T(n,k) = 0 if k < 0 or if k > n.

%F G.f.: (1-(1-2*y)*x)/(1-(1+y)*x+y*(1-2*y)*x^2).

%F Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A108411(n+1), A000007(n), A001333(n+1) for x = -1, 0, 1 respectively. (End)

%e First five rows:

%e 1;

%e 0, 3;

%e 0, 2, 5;

%e 0, 2, 4, 11;

%e 0, 2, 4, 14, 21;

%e First five polynomials u(n,x):

%e 1

%e 3x

%e 2x + 5x^2

%e 2x + 4x^2 + 11x^3

%e 2x + 4x^2 + 14x^3 + 21x^4.

%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_] := 2 x*u[n - 1, x] + 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[%] (* A208328 *)

%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[%] (* A208329 *)

%Y Cf. A208328.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Feb 26 2012