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A140956
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Triangle read by rows: coefficients of the alternating factorial polynomial (x+1)(x-2)(x+3)(x-4)...(x+n*(-1)^(n-1)).
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0
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1, 1, 1, -2, -1, 1, -6, -5, 2, 1, 24, 14, -13, -2, 1, 120, 94, -51, -23, 3, 1, -720, -444, 400, 87, -41, -3, 1, -5040, -3828, 2356, 1009, -200, -62, 4, 1, 40320, 25584, -22676, -5716, 2609, 296, -94, -4, 1, 362880, 270576, -178500, -74120, 17765, 5273, -550, -130, 5, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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).
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REFERENCES
| W. Dunham, Euler The Master of Us All, The Mathematical Association of America (1999), pp. 39-60
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FORMULA
| (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.
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EXAMPLE
| 3A1 =(-1)^2*3*2A1+2A0 =1*3*-1+-2 = -5
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CROSSREFS
| Sequence in context: A047920 A144655 A199063 * A166919 A168641 A143185
Adjacent sequences: A140953 A140954 A140955 * A140957 A140958 A140959
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KEYWORD
| easy,sign,tabl
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AUTHOR
| Ken Grant (plkl(AT)ozemail.com.au), Jul 26 2008
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