A209422 - OEIS

%I #18 Jan 05 2018 02:55:46

%S 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 109,109,82,49,22,10,2,1,177,206,173,112,61,25,11,2,1,287,382,352,252,

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

%C For a discussion and guide to related arrays, see A208510.

%H G. C. Greubel, <a href="/A209422/b209422.txt">Table of n, a(n) for the first 100 rows, flattened</a>

%F u(n,x) = x*u(n-1,x) + v(n-1,x),

%F v(n,x) = u(n-1,x) + v(n-1,x) + 1,

%F where u(1,x)=1, v(1,x)=1.

%F 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 First five rows:

%e    1;

%e    3;

%e    5, 1;

%e    9, 2, 1;

%e   15, 6, 2, 1;

%e First three polynomials v(n,x): 1, 3, 5 + x.

%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_] := u[n - 1, x] + v[n - 1, x] + 1;

%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[%]    (* A209421 *)

%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[%]    (* A209422 *)

%t 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 Cf. A209421, A208510.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Mar 09 2012