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Search: id:A097418
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| A097418 |
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Triangle of coefficients of a certain sequence of polynomials f_n(x) arising in connection with deformations of coordinate rings of type D Kleinian singularities. |
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2
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| 1, 2, 0, 3, 2, 0, 4, 8, 8, 0, 5, 20, 56, 56, 0, 6, 40, 216, 608, 608, 0, 7, 70, 616, 3352, 9440, 9440, 0, 8, 112, 1456, 12928, 70400, 198272, 198272, 0, 9, 168, 3024, 39696, 352768, 1921152, 5410688, 5410668, 0, 10, 240, 5712, 103872, 1364800, 12129664, 66057856
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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f_n(x) has the property that whenever (a,b) is a pair of complex numbers satisfying 2ab = a^2 + 2a + b^2 + 2b, we have f_n(a) + f_n(b) = 2(a^n - b^n)/(a-b) (interpreted as 2na^(n-1) if a=b). Using the pairs (0,0), (0,-2), (-2,-6), (-6,-12), (-12,-20)... this enables us to successively deduce the values of f_n(0), f_n(-2),... (and this of course determines f_n(x)). There may be no simpler characterization.
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REFERENCES
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P. Boddington, Ph.D. thesis, University of Warwick (anticipated 2005).
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EXAMPLE
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The array begins
1
2 0
3 2 0
4 8 8 0
corresponding to the polynomials f_1(x) = 1, f_2(x) = 2x, f_3(x) = 3x^2 + 2x, f_4(x) = 4x^3 + 8x^2 + 8x.
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CROSSREFS
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Sequence in context: A115241 A154559 A143324 this_sequence A154752 A156776 A026728
Adjacent sequences: A097415 A097416 A097417 this_sequence A097419 A097420 A097421
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul Boddington (psb(AT)maths.warwick.ac.uk), Aug 20 2004
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