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Triangle of coefficients of polynomials u(n,x) jointly generated with A208613; see the Formula section.
3

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

%S 1,1,1,1,4,2,1,9,9,3,1,16,26,19,5,1,25,60,70,38,8,1,36,120,197,167,73,

%T 13,1,49,217,469,553,375,137,21,1,64,364,994,1528,1427,803,252,34,1,

%U 81,576,1932,3714,4476,3449,1661,457,55,1,100,870,3510,8196

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208613; see the Formula section.

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

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

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

%e First five rows:

%e 1

%e 1...1

%e 1...4....2

%e 1...9....9....3

%e 1...16...26...19...5

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

%e 1

%e 1 + x

%e 1 + 4x + 2x^2

%e 1 + 9x + 9x^2 + 3x^3

%e 1 + 16x + 26x^2 + 19x^3 + 5x^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_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%] (* A208612 *)

%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[%] (* A208613 *)

%Y Cf. A208613.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Mar 01 2012