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A210875 Triangular array U(n,k) of coefficients of polynomials defined in Comments. 4

%I

%S 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 15,19,27,35,44,37,21,9,17,22,32,43,57,71,60,34,10,19,25,37,51,70,92,

%U 115,97,55,11,21,28,42,59,83,113,149,186,157,89,12,23,31,47,67

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

%C 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 u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).

%C 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 Alternating row sums: 1,0,1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)

%F Column k consists of the partial sums of the following sequence: F(k), 3F(k-1), F(k+2), F(k+1), F(k+1),..., where F=000045 (Fibonacci numbers. That is, U(n+1,k)-U(n,k)=F(k+1) for n>2.

%e First six rows:

%e 1

%e 1...3

%e 3...4....2

%e 4...7....5....3

%e 5...9....10...9....5

%e 6...11...13...17...14...8

%e First three polynomials u(n,x): 1, 1 + 3x, 3 + 4x + 2x^2.

%t u[1, x_] := 1; u[2, x_] := x + 1; z = 14;

%t u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A210875 *)

%Y Cf. A208510, A210881, A210874.

%K nonn,tabl

%O 1,4

%A _Clark Kimberling_, Mar 30 2012

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Last modified February 26 05:51 EST 2020. Contains 332277 sequences. (Running on oeis4.)