

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
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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 HeleShaw Cells, (Advances in Mathematical Fluid Mechanics), Birkhäuser Verlag, 2006, p. 202.


LINKS



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 (* JeanFrançois Alcover, Dec 24 2013 *)


CROSSREFS



KEYWORD

easy,sign,tabf


AUTHOR



EXTENSIONS

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


STATUS

approved



