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Triangle of coefficients of polynomials v(n,x) jointly generated with A209745; see the Formula section.
3

%I #17 Jan 24 2020 03:28:28

%S 1,2,2,3,7,4,5,17,20,8,8,37,65,52,16,13,75,176,210,128,32,21,146,428,

%T 679,616,304,64,34,276,971,1921,2312,1696,704,128,55,511,2097,4970,

%U 7449,7240,4464,1600,256,89,931,4366,12056,21622,26146,21344

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209745; see the Formula section.

%C Row n begins with F(n+1) and ends with 2^(n-1), where F=A000045 (Fibonacci numbers).

%C Alternating row sums: 1,0,0,0,0,0,0,0,0,0,...

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

%C Riordan array ((1+x)/(1-x-x^2), (2x+x^2)/(1-x-x^2)). - _Philippe Deléham_, Mar 24 2012

%C Triangle given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 26 2012

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

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

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

%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(1,0) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k >= 0. - _Philippe Deléham_, Mar 24 2012

%e First five rows:

%e 1;

%e 2, 2;

%e 3, 7, 4;

%e 5, 17, 20, 8;

%e 8, 37, 65, 52, 16;

%e First three polynomials v(n,x):

%e 1

%e 2 + 2x

%e 3 + 7x + 4x^2.

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];

%t v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];

%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[%] (* A209745 *)

%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[%] (* A209746 *)

%Y Cf. A000045, A000079, A209745, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 13 2012