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A145900
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Coefficients of a normalized Schwarzian derivative generating the Neretin polynomials: S(f) = (x^2/6) { D^2 ln(f(x)) - (1/2) [D ln(f(x))]^2 }.
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0
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1, -1, 4, -8, 4, 10, -20, -12, 34, -12, 20, -40, -52, 72, 84, -116, 32, 35, -70, -95, -52, 130, 328, 63, -224, -387, 352, -80, 56, -112, -156, -180, 212, 560, 304, 348, -380, -1416, -540, 640, 1464, -992, 192
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,3
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COMMENTS
| The array contains the coefficients for a normalized Schwarzian: Schw(g(x)) = S(f) = (x^2/6) { D^2 ln(f(x)) - (1/2) [D ln(f(x))]^2} with f(x)= g'(x) = 1 / [1 - c(.) x]^2 = 1 + 2 c(1) x + 3 c(2) x^2 + ....
S(f(x)) = P(2,c) x^2 + P(3,c) x^3 + P(4,c) x^4 + ..., where P(n,c) are the Neretin polynomials with an additional factor of 2.
For proof of integrality of coefficients see MathOverflow link.
Coefficients of P(n,c) sum to zero. - Tom Copeland, Jan 29 2012
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REFERENCES
| B. Gustaffson and A. Vasil'ev, Conformal and Potential Analysis in Hele-Shaw Cells, (Advaces in Mathematical Fluid Mechanics), Birkhauser Verlag, 2006, pg. 202
R. Hidalgo, I. Markina, A. Vasil'ev, Finite dimensional grading of the Virasoro algebra, Georg. Math. J. 14 (2007), 419-434.
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LINKS
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
R. Hidalgo, I. Markina and A. Vasil'ev, Finite dimensional grading of the Virasoro algebra
A. Kirillov, Geometric approach to discrete series of unireps for Vir.
MathOverflow, Conjecture: "Neretin polynomials" for a normalized Schwarzian have integer coefficients
V. Ovsienko and S. Tabachnikov, What is the Schwarzian Derivative?
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FORMULA
| See references for recurrences and lowering operators.
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EXAMPLE
| .. P(0,c) = 0
.. P(1,c) = 0
.. P(2,c) = c(2) - c(1)^2
.. P(3,c) = 4 c(3) - 8 c(2)c(1) + 4 c(1)^3 = 4 3' - 8 2'1' + 4 1'^3
.. P(4,c) = 10 4' - 20 3'1' - 12 2'^2 + 34 2'1'^2 - 12 1'^4
.. P(5,c) = 20 5' - 40 1'4' - 52 2'3' + 72 3'1'^2 + 84 2'^2 1'- 116 2'1'^3 + 32 1'^5
The partitions are arranged in the order of those of Abramowitz and Stegun on pg. 831.
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CROSSREFS
| Sequence in context: A165267 A092159 A141402 * A010298 A196177 A059159
Adjacent sequences: A145897 A145898 A145899 * A145901 A145902 A145903
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KEYWORD
| easy,sign,tabf
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AUTHOR
| Tom Copeland (tcjpn(AT)msn.com), Oct 22 2008
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EXTENSIONS
| Clarified relations among g(x), f(x), and Schwarzian derivative Tom Copeland (tcjpn(AT)msn.com), Dec 08 2009
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