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A210875 Triangular array U(n,k) of coefficients of polynomials defined in Comments.

%I A210875 #10 Oct 26 2024 04:57:13
%S A210875 1,1,1,3,4,2,4,7,5,3,5,9,10,9,5,6,11,13,17,14,8,7,13,16,22,27,23,13,8,
%T A210875 15,19,27,35,44,37,21,9,17,22,32,43,57,71,60,34,10,19,25,37,51,70,92,
%U A210875 115,97,55,11,21,28,42,59,83,113,149,186,157,89,12,23,31,47,67
%N A210875 Triangular array U(n,k) of coefficients of polynomials defined in Comments.
%C A210875 Polynomials u(n,k) are defined by u(n,x)=x*u(n-1,x)+(x^2)*u(n-2,x)+n*(x+1), where u(1)=1 and u(2,x)=x+1.  The array (U(n,k)) is defined by rows:
%C A210875 u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).
%C A210875 In each column, the first number is a Fibonacci number and, with one exception, the difference between each two consecutive terms is a Fibonacci number (see the Formula section).
%C A210875 Alternating row sums: 1,0,1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)
%F A210875 Column k consists of the partial sums of the following sequence: F(k), 3*F(k-1), F(k+2), F(k+1), F(k+1),..., where F=A000045 (Fibonacci numbers). That is, U(n+1,k)-U(n,k)=F(k+1) for n>2.
%e A210875 First six rows:
%e A210875   1
%e A210875   1...1
%e A210875   3...4....2
%e A210875   4...7....5....3
%e A210875   5...9....10...9....5
%e A210875   6...11...13...17...14...8
%e A210875 First three polynomials u(n,x): 1, 1 + 3x, 3 + 4x + 2x^2.
%t A210875 u[1, x_] := 1; u[2, x_] := x + 1; z = 14;
%t A210875 u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);
%t A210875 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210875 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210875 TableForm[cu]
%t A210875 Flatten[%]   (* A210875 *)
%Y A210875 Cf. A208510, A210881, A210874.
%K A210875 nonn,tabl
%O A210875 1,4
%A A210875 _Clark Kimberling_, Mar 30 2012