1,4

The coefficients belong to the rows of the following tree which is built up from the following polynomials:

1

(x+1) =x+1

(x+1)(x-2)= x^2-x-2

(x+1)(x-2)(x+3)= x^3+2x^2-5x-6

(x+1)(x-2)(x+3(x-4)= x^4-2x^3-13x^2+14x+24 and so on.

1

1 1

-2 -1 1

-6 -5 2 1

24 14 -13 -2 1

. . . . . .

Every term of the tree can be denoted by nAm as follows:

1A0 1A1

2A0 2A1 2A2

3A0 3A1 3A2 3A3

4A0 4A1 4A2 4A3 4A4

. . . . . .

nA0 nA1 nA2 nA3 nA4 . . nAn

Properties:

(1) For all positive integers n, nAn = 1

(2) nA0 = n!(-1)^T where T=int(n/2)

(3) For terms of the tree, which are not on the border of the tree, nAm = (-1)^(n+1) * n * (n-1)Am + (n-1)A(m-1). E.g. for m=1, nA1 = (-1)^(n+1) * n * (n-1)A1 + (n-1)A0 and this term has coordinates (n,m)=(n,1). The term with co-ordinates (3,1) = 3A1 =(-1)^2*3*2A1+2A0 =1*3*-1+-2 = -5

(4) All terms of the tree are integers; this follows from (1), (2) and (3).

(5) For n >= 2, the sum of the terms of any row is given by: Sum of terms of row n = (-1)^T*[n+(-1)^(n-1)]*Sum of terms of row (n-1).

(6) For all n, nA0/[(-1)^T * n! ] = 1

(7) As n approaches infinity, (nA2) / ((-1)^T * n!) approaches (1/2) * (ln 2)^2 - (pi^2) / 12.

As n approaches infinity, (nA3) / ((-1)^T * n!) approaches (1/6) * (ln 2)^3 - (1/12) * (pi^2) * (ln 2) + (1/4) * zeta(3) where zeta(3) = sum_{m=1 .. infinity} 1/m^3.

(8) Since all terms of the tree are integers, then it follows from (7) that Zeta(3) = (1/3) * (pi^2) * (ln2) - (2/3) * (ln 2)^3 + C, where C is a rational number.

Remember that T equals the integer part of (n divided by 2) and n! equals (factorial n).

W. Dunham, Euler The Master of Us All, The Mathematical Association of America (1999), pp. 39-60

(1) For all positive integers n, nAn = 1. (2) nA0 = n!(-1)^T where T=int(n/2). (3) For terms of the tree, which are not on the border of the tree, nAm = (-1)^(n+1) * n * (n-1)Am + (n-1)A(m-1). E.g. for m=1, nA1 = (-1)^(n+1) * n * (n-1)A1 + (n-1)A0.

3A1 =(-1)^2*3*2A1+2A0 =1*3*-1+-2 = -5

Sequence in context: A135899 A047920 A144655 this_sequence A166919 A168641 A143185

Adjacent sequences: A140953 A140954 A140955 this_sequence A140957 A140958 A140959

easy,sign,tabl

Ken Grant (plkl(AT)ozemail.com.au), Jul 26 2008