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A210866 Triangle of coefficients of polynomials u(n,x) jointly generated with A210867; see the Formula section.

%I A210866 #7 Jul 12 2012 00:40:00
%S A210866 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 A210866 8,1,28,231,700,867,503,134,13,1,36,406,1792,3332,2914,1279,268,21,1,
%U A210866 45,666,4074,10955,13882,9084,3122,527,34,1,55,1035,8430,31563
%N A210866 Triangle of coefficients of polynomials u(n,x) jointly generated with A210867; see the Formula section.
%C A210866 Row n starts with 1, followed by the n-th triangular number, and ends with the n-th Fibonacci number.
%C A210866 For a discussion and guide to related arrays, see A208510.
%F A210866 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A210866 v(n,x)=(x+n)*u(n-1,x)+x*v(n-1,x)-x,
%F A210866 where u(1,x)=1, v(1,x)=1.
%e A210866 First five rows:
%e A210866 1
%e A210866 1...1
%e A210866 1...3....1
%e A210866 1...6....6....2
%e A210866 1...10...21...14...3
%e A210866 First three polynomials u(n,x): 1, 1 + x, 1 + 3x + x^2.
%t A210866 u[1, x_] := 1; v[1, x_] := 1; z = 14;
%t A210866 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A210866 v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - x;
%t A210866 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210866 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210866 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210866 TableForm[cu]
%t A210866 Flatten[%]   (* A210866 *)
%t A210866 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210866 TableForm[cv]
%t A210866 Flatten[%]   (* A210867 *)
%Y A210866 Cf. A210867, A208510.
%K A210866 nonn,tabl
%O A210866 1,5
%A A210866 _Clark Kimberling_, Mar 29 2012