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A209422 Triangle of coefficients of polynomials v(n,x) jointly generated with A209415; see the Formula section.

%I A209422 #18 Jan 05 2018 02:55:46
%S A209422 1,3,5,1,9,2,1,15,6,2,1,25,13,7,2,1,41,28,16,8,2,1,67,56,38,19,9,2,1,
%T A209422 109,109,82,49,22,10,2,1,177,206,173,112,61,25,11,2,1,287,382,352,252,
%U A209422 146,74,28,12,2,1,465,697,701,543,347,184,88,31,13,2,1,753,1256,1368,1144,784,459,226,103,34,14,2
%N A209422 Triangle of coefficients of polynomials v(n,x) jointly generated with A209415; see the Formula section.
%C A209422 For a discussion and guide to related arrays, see A208510.
%H A209422 G. C. Greubel, <a href="/A209422/b209422.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F A209422 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209422 v(n,x) = u(n-1,x) + v(n-1,x) + 1,
%F A209422 where u(1,x)=1, v(1,x)=1.
%F A209422 G.f.: (1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)) = 1 + 3*t + (5 + x)*t^2 + ... . - _G. C. Greubel_, Jan 03 2018
%e A209422 First five rows:
%e A209422    1;
%e A209422    3;
%e A209422    5, 1;
%e A209422    9, 2, 1;
%e A209422   15, 6, 2, 1;
%e A209422 First three polynomials v(n,x): 1, 3, 5 + x.
%t A209422 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209422 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209422 v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A209422 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209422 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209422 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209422 TableForm[cu]
%t A209422 Flatten[%]    (* A209421 *)
%t A209422 Table[Expand[v[n, x]], {n, 1, z}]
%t A209422 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209422 TableForm[cv]
%t A209422 Flatten[%]    (* A209422 *)
%t A209422 CoefficientList[CoefficientList[Series[(1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)), {t, 0, 10}], t], x]// Flatten (* _G. C. Greubel_, Jan 03 2018 *)
%Y A209422 Cf. A209421, A208510.
%K A209422 nonn,tabf
%O A209422 1,2
%A A209422 _Clark Kimberling_, Mar 09 2012