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A209413 Triangle of coefficients of polynomials v(n,x) jointly generated with A209172; see the Formula section.

%I A209413 #29 Oct 26 2024 04:21:35
%S A209413 1,1,2,1,3,4,1,5,7,8,1,6,17,15,16,1,8,23,49,31,32,1,9,39,72,129,63,64,
%T A209413 1,11,48,150,201,321,127,128,1,12,70,198,501,522,769,255,256,1,14,82,
%U A209413 338,699,1524,1291,1793,511,512,1,15,110,420,1375,2223,4339,3084,4097,1023,1024
%N A209413 Triangle of coefficients of polynomials v(n,x) jointly generated with A209172; see the Formula section.
%C A209413 For n > 1, n-th alternating row sum = (-1)^(n-1)*F(2*n-3), where F=A000045 (Fibonacci numbers).
%C A209413 Coefficient of x^(n-1) in u(n,x): 2^(n-1).
%C A209413 For a discussion and guide to related arrays, see A208510.
%C A209413 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, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 11 2012
%H A209413 G. C. Greubel, <a href="/A209413/b209413.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A209413 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209413 v(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A209413 where u(1,x)=1, v(1,x)=1.
%F A209413 From _Philippe Deléham_, Mar 11 2012: (Start)
%F A209413 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209413 T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(2,0) = 1, T(2,1) = 2, T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.
%F A209413 G.f.: (1+x-3*y*x-y*x^2+2*y^2*x^2)/(1-3*y*x-(1-2y^2)*x^2). (End)
%e A209413 First five rows:
%e A209413   1;
%e A209413   1,  2;
%e A209413   1,  3,  4;
%e A209413   1,  5,  7,  8;
%e A209413   1,  6, 17, 15, 16;
%e A209413 First three polynomials v(n,x):
%e A209413   1
%e A209413   1 + 2x
%e A209413   1 + 3x + 4x^2.
%e A209413 From _Philippe Deléham_, Mar 11 2012: (Start)
%e A209413 (1, 0, -1/2, -1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, ...) begins:
%e A209413   1;
%e A209413   1,   0;
%e A209413   1,   2,   0;
%e A209413   1,   3,   4,   0;
%e A209413   1,   5,   7,   8,   0;
%e A209413   1,   6,  17,  15,  16,   0;
%e A209413   1,   8,  23,  49,  31,  32,   0;
%e A209413   1,   9,  39,  72, 129,  63,  64,   0;
%e A209413   1,  11,  48, 150, 201, 321, 127, 128,   0; (End)
%t A209413 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209413 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209413 v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A209413 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209413 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209413 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209413 TableForm[cu]
%t A209413 Flatten[%]    (* A209172 *)
%t A209413 Table[Expand[v[n, x]], {n, 1, z}]
%t A209413 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209413 TableForm[cv]
%t A209413 Flatten[%]    (* A209413 *)
%t A209413 CoefficientList[CoefficientList[Series[(1 + x - 3*y*x - y*x^2 + 2*y^2*x^2)/(1 - 3*y*x - (1 - 2 y^2)*x^2), {x,0,10}, {y,0,10}], x],y] // Flatten (* _G. C. Greubel_, Jan 03 2018 *)
%Y A209413 Cf. A209172, A208510.
%K A209413 nonn,tabl
%O A209413 1,3
%A A209413 _Clark Kimberling_, Mar 08 2012