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%I #20 Jan 24 2020 03:29:27
%S 1,1,2,1,3,3,1,5,7,4,1,6,15,14,5,1,8,23,36,25,6,1,9,36,69,76,41,7,1,
%T 11,48,123,176,147,63,8,1,12,66,192,355,400,266,92,9,1,14,82,292,635,
%U 910,834,456,129,10,1,15,105,410,1065,1833,2131,1626,747,175,11
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209415; see the Formula section.
%C Alternating row sums: 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 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_, Apr 01 2012
%H G. C. Greubel, <a href="/A209416/b209416.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F u(n,x) = x*u(n-1,x) + v(n-1,x),
%F v(n,x) = (x+1)*u(n-1,x) + x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Apr 01 2012: (Start)
%F As DELTA-triangle T(n,k) with 0 <= k <=n:
%F G.f.: (1+x-2*y*x-y*x^2+y^2*x^2)/((1-2*y*x-x^2-y*x^2+y^2*x^2).
%F T(n,k) = 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e First five rows:
%e 1;
%e 1, 2;
%e 1, 3, 3;
%e 1, 5, 7, 4;
%e 1, 6, 15, 14, 5;
%e First three polynomials v(n,x):
%e 1
%e 1 + 2x
%e 1 + 3x + 3x^2.
%e From _Philippe Deléham_, Apr 01 2012: (Start)
%e (1, 0, -1/2, -1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 3, 3, 0;
%e 1, 5, 7, 4, 0;
%e 1, 6, 15, 14, 5, 0;
%e 1, 8, 23, 36, 25, 6, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t v[n_, x_] := (x + 1)*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[%] (* A209415 *)
%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[%] (* A209416 *)
%t CoefficientList[CoefficientList[Series[(1 + x - 2*y*x - y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2), {x,0,10}, {y,0,10}], x], y] // Flatten (* _G. C. Greubel_, Jan 03 2018 *)
%Y Cf. A209416, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 09 2012
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