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%I #14 Jan 24 2020 03:27:20
%S 1,1,2,1,5,3,1,9,12,5,1,14,31,27,8,1,20,65,89,55,13,1,27,120,230,222,
%T 108,21,1,35,203,511,684,514,205,34,1,44,322,1022,1777,1834,1125,381,
%U 55,1,54,486,1890,4095,5442,4563,2367,696,89,1,65,705,3288,8625
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A102756; see the Formula section.
%C Top edge: (1,2,3,5,8,...) = A000045(n+1), Fibonacci numbers.
%C Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,...
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle T(n,k) given by (1, 0, 1/2, 1/2, 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 08 2012
%F u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F v(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 08 2012: (Start)
%F As DELTA-triangle T(n,k) with 0 <= k <= n:
%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
%F G.f.: (1-x-y*x+y*x^2-y^2*x^2)/(1-(2+y)*x-(y^2-1)*x^2).
%F Sum_{k=0..n, n>=1} T(n,k)*x^k = A153881(n), A000012(n), A000244(n-1), A126473(n-1) for x = -1, 0, 1, 2 respectively. (End)
%e First five rows:
%e 1;
%e 1, 2;
%e 1, 5, 3;
%e 1, 9, 12, 5;
%e 1, 14, 31, 27, 8;
%e First three polynomials v(n,x):
%e 1
%e 1 + 2x
%e 1 + 5x + 3x^2.
%e From _Philippe Deléham_, Mar 08 2012: (Start)
%e (1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0...) begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 5, 3, 0;
%e 1, 9, 12, 5, 0;
%e 1, 14, 31, 27, 8, 0;
%e 1, 20, 65, 89, 55, 13, 0; ...
%e with row sums 1, 1, 3, 9, 27, 81, 243, 729, ... (powers of 3). (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + (x + 1)*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[%] (* A102756 *)
%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[%] (* A209130 *)
%Y Cf. A102756, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 05 2012
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