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

%I A209420 #18 Jan 04 2018 01:35:12
%S A209420 1,2,2,3,6,3,5,14,13,4,8,30,41,24,5,13,60,109,96,40,6,21,116,262,308,
%T A209420 196,62,7,34,218,590,868,743,364,91,8,55,402,1267,2240,2413,1604,630,
%U A209420 128,9,89,730,2627,5424,7046,5926,3186,1032,174,10,144,1310,5299,12516,19040,19382,13255,5928,1617,230
%N A209420 Triangle of coefficients of polynomials v(n,x) jointly generated with A209419; see the Formula section.
%C A209420 Column 1:  Fibonacci numbers (A000045).
%C A209420 For a discussion and guide to related arrays, see A208510.
%C A209420 Triangle given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 26 2012
%H A209420 G. C. Greubel, <a href="/A209420/b209420.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A209420 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209420 v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x),
%F A209420 where u(1,x)=1, v(1,x)=1.
%F A209420 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) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 26 2012
%F A209420 G.f.: x*(1 + x)/(1 - x - x^2 - 2*y*x + y^2*x^2). - _G. C. Greubel_, Jan 03 2018
%e A209420 First five rows:
%e A209420   1;
%e A209420   2,  2;
%e A209420   3,  6,  3;
%e A209420   5, 14, 13,  4;
%e A209420   8, 30, 41, 24,  5;
%e A209420 First three polynomials v(n,x): 1, 2 + 2x, 3 + 6x + 3x^2.
%t A209420 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209420 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209420 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209420 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209420 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209420 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209420 TableForm[cu]
%t A209420 Flatten[%]    (* A209419 *)
%t A209420 Table[Expand[v[n, x]], {n, 1, z}]
%t A209420 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209420 TableForm[cv]
%t A209420 Flatten[%]    (* A209420 *)
%t A209420 CoefficientList[CoefficientList[Series[x*(1 + x)/(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 A209420 Cf. A209419, A208510.
%K A209420 nonn,tabl
%O A209420 1,2
%A A209420 _Clark Kimberling_, Mar 09 2012