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Triangle of coefficients of polynomials u(n,x) jointly generated with A210867; see the Formula section.
4

%I #7 Jul 12 2012 00:40:00

%S 1,1,1,1,3,1,1,6,6,2,1,10,21,14,3,1,15,55,65,31,5,1,21,120,235,187,65,

%T 8,1,28,231,700,867,503,134,13,1,36,406,1792,3332,2914,1279,268,21,1,

%U 45,666,4074,10955,13882,9084,3122,527,34,1,55,1035,8430,31563

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210867; see the Formula section.

%C Row n starts with 1, followed by the n-th triangular number, and ends with the n-th Fibonacci number.

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

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

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

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

%e First five rows:

%e 1

%e 1...1

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

%e 1...6....6....2

%e 1...10...21...14...3

%e First three polynomials u(n,x): 1, 1 + x, 1 + 3x + x^2.

%t u[1, x_] := 1; v[1, x_] := 1; z = 14;

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

%t v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - 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[%] (* A210866 *)

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A210867 *)

%Y Cf. A210867, A208510.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Mar 29 2012