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 A145900 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 }. 1

%I

%S 1,-1,4,-8,4,10,-20,-12,34,-12,20,-40,-52,72,84,-116,32,35,-70,-95,

%T -52,130,328,63,-224,-387,352,-80,56,-112,-156,-180,212,560,304,348,

%U -380,-1416,-540,640,1464,-992,192

%N 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 }.

%C 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 + ....

%C 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.

%C For proof of integrality of coefficients see MathOverflow link.

%C Coefficients of P(n,c) sum to zero. - _Tom Copeland_, Jan 29 2012

%D 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.

%D B. Gustaffson and A. Vasil'ev, Conformal and Potential Analysis in Hele-Shaw Cells, (Advances in Mathematical Fluid Mechanics), Birkhauser Verlag, 2006, p. 202.

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.

%H Tom Copeland, <a href="https://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera</a>.

%H R. Hidalgo, I. Markina and A. Vasil'ev, <a href="http://www.uib.no/People/ima083/bibliogr_files/HidMarVas.pdf">Finite dimensional grading of the Virasoro algebra</a>, Georg. Math. J. 14 (2007), 419-434.

%H A. Kirillov, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.260">Geometric approach to discrete series of unireps for Vir.</a>

%H I. Markina, D. Prokhorov, and A. Vasil'ev, <a href="https://arxiv.org/abs/math/0608532">Sub-Riemannian geometry of the coefficients of univalent functions</a>, arXiv:math/0608532 [math.CV], p. 11, 2006.

%H MathOverflow, <a href="http://mathoverflow.net/questions/85871">Conjecture: "Neretin polynomials" for a normalized Schwarzian have integer coefficients</a>

%H V. Ovsienko and S. Tabachnikov, <a href="http://www.ams.org/notices/200901/tx090100034p.pdf">What is the Schwarzian Derivative?</a>, AMS Notices 56 (01), 34-36.

%H A. Vasilâ€™ev, <a href="https://arxiv.org/abs/math-ph/0509072">Energy characteristics of subordination chains</a>, arXiv:math-ph/0509072 [math-ph], p. 11, 2005.

%F See references for recurrences and lowering operators.

%e .. P(0,c) = 0

%e .. P(1,c) = 0

%e .. P(2,c) = c(2) - c(1)^2

%e .. P(3,c) = 4 c(3) - 8 c(2)c(1) + 4 c(1)^3 = 4 3' - 8 2'1' + 4 1'^3

%e .. P(4,c) = 10 4' - 20 3'1' - 12 2'^2 + 34 2'1'^2 - 12 1'^4

%e .. 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

%e The partitions are arranged in the order of those of Abramowitz and Stegun on p. 831.

%t 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-FranÃ§ois Alcover_, Dec 24 2013 *)

%K easy,sign,tabf

%O 2,3

%A _Tom Copeland_, Oct 22 2008

%E Clarified relations among g(x), f(x), and Schwarzian derivative _Tom Copeland_, Dec 08 2009

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