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A209419 Triangle of coefficients of polynomials u(n,x) jointly generated with A209420; see the Formula section.

%I A209419 #17 Jan 04 2018 01:35:04
%S A209419 1,1,1,2,3,1,3,8,6,1,5,17,21,10,1,8,35,58,45,15,1,13,68,144,154,85,21,
%T A209419 1,21,129,330,452,350,147,28,1,34,239,719,1198,1195,714,238,36,1,55,
%U A209419 436,1506,2959,3611,2799,1344,366,45,1,89,785,3063,6930,10005,9537,5985,2376,540,55,1
%N A209419 Triangle of coefficients of polynomials u(n,x) jointly generated with A209420; see the Formula section.
%C A209419 Column 1:  Fibonacci numbers (A000045)
%C A209419 Alternating row sums: (1,0,0,0,0,0,0,0,0,0,0,0,...)
%C A209419 For a discussion and guide to related arrays, see A208510.
%C A209419 Triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 26 2012
%H A209419 G. C. Greubel, <a href="/A209419/b209419.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A209419 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209419 v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x),
%F A209419 where u(1,x)=1, v(1,x)=1.
%F A209419 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 26 2012
%F A209419 G.f.: x*(1 - x*y)/(1 - x - x^2 - 2*y*x + y^2*x^2). - _G. C. Greubel_, Jan 03 2018
%e A209419 First five rows:
%e A209419   1;
%e A209419   1,  1;
%e A209419   2,  3,  1;
%e A209419   3,  8,  6,  1;
%e A209419   5, 17, 21, 10,  1;
%e A209419 First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.
%t A209419 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209419 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209419 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209419 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209419 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209419 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209419 TableForm[cu]
%t A209419 Flatten[%]    (* A209419 *)
%t A209419 Table[Expand[v[n, x]], {n, 1, z}]
%t A209419 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209419 TableForm[cv]
%t A209419 Flatten[%]    (* A209420 *)
%t A209419 CoefficientList[CoefficientList[Series[(1*x - x^2*y)/(1 - x - x^2 - 2*y*x + y^2*x^2), {x,0,10}, {y,0,10}], x], y] // Flatten (* _G. C. Greubel_, Jan 03 2018 *)
%Y A209419 Cf. A209420, A208510.
%K A209419 nonn,tabl
%O A209419 1,4
%A A209419 _Clark Kimberling_, Mar 09 2012