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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 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, Georg. Math. J. 14 (2007), 419-434. 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. 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.)