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A209764 Triangle of coefficients of polynomials v(n,x) jointly generated with A209763; see the Formula section. 3

%I

%S 1,2,2,3,4,4,4,8,14,8,5,14,32,34,16,6,22,62,96,86,32,7,32,108,218,280,

%T 202,64,8,44,174,432,718,760,470,128,9,58,264,778,1584,2194,1992,1066,

%U 256,10,74,382,1304,3142,5360,6382,5048,2390,512,11,92,532

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209763; see the Formula section.

%C Row n begins with n and ends with 2^(n-1).

%C Row sums: 1,4,11,34,101,304,911,2734,... A060925.

%C Alternating row sums: 1,0,3,2,5,4,7,6,... A060925.

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

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

%F v(n,x)=2x*u(n-1,x)+v(n-1,x)+1,

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

%e First five rows:

%e 1

%e 2...2

%e 3...4....4

%e 4...8....14...8

%e 5...14...32...34...16

%e First three polynomials v(n,x): 1, 2 + 2x , 3 + 4x + 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_] := 2 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[%] (* A209763 *)

%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[%] (* A209764 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A081250 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A060925 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442*)

%Y Cf. A209663, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 14 2012

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Last modified November 19 03:27 EST 2019. Contains 329310 sequences. (Running on oeis4.)