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A210874
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Triangular array U(n,k) of coefficients of polynomials defined in Comments.
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4
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1, 2, 3, 3, 5, 4, 4, 7, 7, 7, 5, 9, 10, 12, 11, 6, 11, 13, 17, 19, 18, 7, 13, 16, 22, 27, 31, 29, 8, 15, 19, 27, 35, 44, 50, 47, 9, 17, 22, 32, 43, 57, 71, 81, 76, 10, 19, 25, 37, 51, 70, 92, 115, 131, 123, 11, 21, 28, 42, 59, 83, 113, 149, 186, 212, 199, 12, 23, 31
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OFFSET
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1,2
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COMMENTS
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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)=3x+2. The array (U(n,k)) is defined by rows:
u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).
In each column, the first number is a Lucas number and the difference between each two consecutive terms is a Fibonacci number (see the Formula section).
Alternating row sums: 1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)
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LINKS
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FORMULA
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Column k consists of the partial sums of the following sequence: L(k), F(k+1), F(k+1), F(k+1), F(k+1),..., where L=A000032 (Lucas numbers) and F=000045 (Fibonacci numbers. That is, U(n+1,k)-U(n,k)=F(k+1) for n>1.
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EXAMPLE
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First six rows:
1
2...3
3...5...4
4...7...7.....7
5...9...10....12...11
6...11...13...17...19...18
First three polynomials u(n,x): 1, 2 + 3x, 3 + 5x + 4x^2.
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MATHEMATICA
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u[1, x_] := 1; u[2, x_] := 3 x + 2; z = 14;
u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);
Table[Expand[u[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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