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Triangle of coefficients of polynomials v(n,x) jointly generated with A208511; see the Formula section.
3

%I #5 Mar 30 2012 18:58:13

%S 1,2,2,2,5,4,2,7,12,8,2,9,21,28,16,2,11,32,58,64,32,2,13,45,101,152,

%T 144,64,2,15,60,159,296,384,320,128,2,17,77,234,513,824,944,704,256,2,

%U 19,96,328,822,1554,2208,2272,1536,512,2,21,117,443,1244,2685

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208511; see the Formula section.

%C Alternating row sums are signed Fibonacci numbers (A000045).

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

%F v(n,x)=u(n-1,x)+2x*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 2...5...4

%e 2...7...12...8

%e 2...9...21...28...16

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

%e 1

%e 2 + 2x

%e 2 + 5x + 4x^2

%e 2 + 7x + 12x^2 + 8x^3

%e 2 + 9x + 21x^2 + 28x^3 + 16x^4

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

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

%t v[n_, x_] := u[n - 1, x] + 2 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[%] (* A208511 *)

%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[%] (* A208512 *)

%Y Cf. A208511.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 28 2012