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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 log(f(x)) - (1/2) [D log(f(x))]^2 }.
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1
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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
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OFFSET
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2,3
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COMMENTS
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The array contains the coefficients for a normalized Schwarzian: Schw(g(x)) = S(f) = (x^2/6) { D^2 log(f(x)) - (1/2) [D log(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
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H. Airault, "Symmetric sums associated to the factorization of Grunsky coefficients," in Groups and Symmetries: From Neolithic Scots to John McKay, CRM Proceedings and Lecture Notes: Vol. 47, edited by J. Harnad and P. Winternitz, American Mathematical Society, p. 5, 2009.
B. Gustaffson and A. Vasil'ev, Conformal and Potential Analysis in Hele-Shaw Cells, (Advances in Mathematical Fluid Mechanics), Birkhäuser Verlag, 2006, p. 202.
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LINKS
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FORMULA
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See references for recurrences and lowering operators.
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EXAMPLE
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.. 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 p. 831.
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MATHEMATICA
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max = 7; f[x_] := 1+Sum[(k+1)*c[k]*x^k, {k, 1, max}]; Lf[x_] := Log[f[x]]; s = (x^2/6)*(Lf''[x]-1/2*Lf'[x]^2); coes = CoefficientList[Series[s, {x, 0, max}], x]; p[n_] := coes[[n+1]]; row[n_] := Module[{r, r1, r2, r3, r4, asteg, pos}, r = List @@ Expand[p[n]]; r1 = r /. c[_] -> 1; r2 = r/r1; r3 = (r2 /. Times -> List /. c[i_]^k_ :> Array[i&, k] ) /. c[i_] :> {i}; r4 = Flatten /@ r3; asteg = Reverse /@ IntegerPartitions[n] //. {a___List, b_List, c_List, d___List} /; Length[b] > Length[c] :> {a, c, b, d}; Do[pos[i] = Position[asteg, r4[[i]], 1, 1][[1, 1]], {i, 1, Length[r]}]; Table[r1[[pos[i]]], {i, 1, Length[r]}]]; Table[row[n], {n, 2, max}] // Flatten (* Jean-François Alcover, Dec 24 2013 *)
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CROSSREFS
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KEYWORD
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easy,sign,tabf
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AUTHOR
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EXTENSIONS
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Clarified relations among g(x), f(x), and Schwarzian derivative Tom Copeland, Dec 08 2009
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STATUS
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approved
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