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A208329 Triangle of coefficients of polynomials v(n,x) jointly generated with A208328; see the Formula section.

%I A208329 #19 Oct 26 2024 03:32:16
%S A208329 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 A208329 2,4,26,56,130,188,171,0,2,4,30,68,186,324,438,341,0,2,4,34,80,250,
%U A208329 492,834,984,683,0,2,4,38,92,322,692,1374,2028,2202,1365,0,2,4,42
%N A208329 Triangle of coefficients of polynomials v(n,x) jointly generated with A208328; see the Formula section.
%C A208329 Row sums, u(n,1):  A000129
%C A208329 Row sums, v(n,1):  A001333
%C A208329 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 A208329 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A208329 v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),
%F A208329 where u(1,x)=1, v(1,x)=1.
%F A208329 From _Philippe Deléham_, Feb 27 2012: (Start)
%F A208329 As triangle T(n,k), 0 <= k <= n:
%F A208329 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 A208329 G.f.: (1-(1-2*y)*x)/(1-(1+y)*x+y*(1-2*y)*x^2).
%F A208329 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 A208329 First five rows:
%e A208329   1;
%e A208329   0,  3;
%e A208329   0,  2,  5;
%e A208329   0,  2,  4, 11;
%e A208329   0,  2,  4, 14, 21;
%e A208329 First five polynomials u(n,x):
%e A208329   1
%e A208329      3x
%e A208329      2x + 5x^2
%e A208329      2x + 4x^2 + 11x^3
%e A208329      2x + 4x^2 + 14x^3 + 21x^4.
%t A208329 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208329 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208329 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
%t A208329 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208329 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208329 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208329 TableForm[cu]
%t A208329 Flatten[%]  (* A208328 *)
%t A208329 Table[Expand[v[n, x]], {n, 1, z}]
%t A208329 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208329 TableForm[cv]
%t A208329 Flatten[%]  (* A208329 *)
%Y A208329 Cf. A208328.
%K A208329 nonn,tabl
%O A208329 1,3
%A A208329 _Clark Kimberling_, Feb 26 2012