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A209830 Triangle of coefficients of polynomials u(n,x) jointly generated with A209831; see the Formula section.

%I A209830 #22 Oct 26 2024 04:21:47
%S A209830 1,1,2,1,5,5,1,7,18,13,1,10,35,59,34,1,12,61,147,185,89,1,15,90,302,
%T A209830 558,564,233,1,17,129,527,1324,1986,1685,610,1,20,170,854,2653,5350,
%U A209830 6761,4957,1597,1,22,222,1278,4811,12066,20383,22277,14406,4181,1
%N A209830 Triangle of coefficients of polynomials u(n,x) jointly generated with A209831; see the Formula section.
%C A209830 Each row begins with 1 and ends with an odd-indexed Fibonacci number.
%C A209830 Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
%C A209830 For a discussion and guide to related arrays, see A208510.
%C A209830 Subtriangle of the triangle given by (1, 0, 1/2, -3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 16 2012
%F A209830 u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
%F A209830 v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F A209830 where u(1,x)=1, v(1,x)=1.
%F A209830 As DELTA-triangle with 0 <= k <= n: G.f.: (1+x-3*y*x-3*y*x^2+y^2*x^2)/(1-3*y*x-x^2-2*y*x^2+y^2*x^2). - _Philippe Deléham_, Mar 16 2012
%F A209830 As DELTA-triangle: T(n,k) = 3*T(n-1,k-1) + T(n-2,k) + 2*T(n-2,k-1) - 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. - _Philippe Deléham_, Mar 16 2012
%e A209830 First five rows:
%e A209830   1;
%e A209830   1,  2;
%e A209830   1,  5,  5;
%e A209830   1,  7, 18, 13;
%e A209830   1, 10, 35, 59, 34;
%e A209830 First three polynomials u(n,x):
%e A209830   1
%e A209830   1 + 2x
%e A209830   1 + 5x + 5x^2.
%e A209830 From _Philippe Deléham_, Mar 16 2012: (Start)
%e A209830 (1, 0, 1/2, -3/2, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 0, 0, ...) begins:
%e A209830   1;
%e A209830   1,  0;
%e A209830   1,  2,  0;
%e A209830   1,  5,  5,  0;
%e A209830   1,  7, 18, 13,  0;
%e A209830   1, 10, 35, 59, 34, 0; (End)
%t A209830 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209830 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209830 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A209830 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209830 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209830 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209830 TableForm[cu]
%t A209830 Flatten[%]    (* A209830 *)
%t A209830 Table[Expand[v[n, x]], {n, 1, z}]
%t A209830 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209830 TableForm[cv]
%t A209830 Flatten[%]    (* A209831 *)
%Y A209830 Cf. A209831, A208510.
%K A209830 nonn,tabl
%O A209830 1,3
%A A209830 _Clark Kimberling_, Mar 13 2012