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A210739 Triangle of coefficients of polynomials u(n,x) jointly generated with A210740; see the Formula section.

%I A210739 #13 Aug 01 2025 07:10:58
%S A210739 1,1,3,1,4,8,1,4,14,21,1,4,15,46,55,1,4,15,55,145,144,1,4,15,56,196,
%T A210739 444,377,1,4,15,56,208,678,1331,987,1,4,15,56,209,764,2282,3926,2584,
%U A210739 1,4,15,56,209,779,2762,7499,11434,6765,1,4,15,56,209,780,2892,9804,24141,32960,17711
%N A210739 Triangle of coefficients of polynomials u(n,x) jointly generated with A210740; see the Formula section.
%C A210739 Rows end with even-indexed Fibonacci numbers.
%C A210739 Limiting row: A001353
%C A210739 Row sums: A003562
%C A210739 Alternating row sums: A000975 (signed)
%C A210739 For a discussion and guide to related arrays, see A208510.
%F A210739 u(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
%F A210739 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A210739 where u(1,x)=1, v(1,x)=1.
%e A210739 First five rows:
%e A210739   1
%e A210739   1   3
%e A210739   1   4   8
%e A210739   1   4   14   21
%e A210739   1   4   15   46   55
%e A210739 First three polynomials u(n,x): 1, 1 + 3*x, 1 + 4*x + 8*x^2.
%t A210739 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210739 u[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210739 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210739 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210739 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210739 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210739 TableForm[cu]
%t A210739 Flatten[%]    (* A210739 *)
%t A210739 Table[Expand[v[n, x]], {n, 1, z}]
%t A210739 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210739 TableForm[cv]
%t A210739 Flatten[%]    (* A210740 *)
%Y A210739 Cf. A210740, A208510.
%K A210739 nonn,tabl
%O A210739 1,3
%A A210739 _Clark Kimberling_, Mar 24 2012