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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
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; text; internal format)
OFFSET

2,3

COMMENTS

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

REFERENCES

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), Birkhauser Verlag, 2006, p. 202.

LINKS

Table of n, a(n) for n=2..44.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.

Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera.

R. Hidalgo, I. Markina and A. Vasil'ev, Finite dimensional grading of the Virasoro algebra, Georg. Math. J. 14 (2007), 419-434.

A. Kirillov, Geometric approach to discrete series of unireps for Vir.

I. Markina, D. Prokhorov, and A. Vasil'ev, Sub-Riemannian geometry of the coefficients of univalent functions, arXiv:math/0608532 [math.CV], p. 11, 2006.

MathOverflow, Conjecture: "Neretin polynomials" for a normalized Schwarzian have integer coefficients

V. Ovsienko and S. Tabachnikov, What is the Schwarzian Derivative?, AMS Notices 56 (01), 34-36.

A. Vasil’ev, Energy characteristics of subordination chains, arXiv:math-ph/0509072 [math-ph], p. 11, 2005.

FORMULA

See references for recurrences and lowering operators.

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 p. 831.

MATHEMATICA

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 *)

CROSSREFS

Sequence in context: A322258 A141402 A276619 * A278676 A010298 A196177

Adjacent sequences:  A145897 A145898 A145899 * A145901 A145902 A145903

KEYWORD

easy,sign,tabf

AUTHOR

Tom Copeland, Oct 22 2008

EXTENSIONS

Clarified relations among g(x), f(x), and Schwarzian derivative Tom Copeland, Dec 08 2009

STATUS

approved

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Last modified December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)