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A263916 Coefficients of the Faber partition polynomials. 21
-1, -2, 1, -3, 3, -1, -4, 4, 2, -4, 1, -5, 5, 5, -5, -5, 5, -1, -6, 6, 6, -6, 3, -12, 6, -2, 9, -6, 1, -7, 7, 7, -7, 7, -14, 7, -7, -7, 21, -7, 7, -14, 7, -1, -8, 8, 8, -8, 8, -16, 8, 4, -16, -8, 24, -8, -8, 12, 24, -32, 8, 2, -16, 20, -8, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The coefficients of the Faber polynomials F(n,b(1),b(2),...,b(n)) (Bouali, p. 52) in the order of the partitions of Abramowitz and Stegun. Compare with A115131 and A210258.

These polynomials occur in discussions of the Virasoro algebra, univalent function spaces and the Schwarzian derivative, symmetric functions, and free probability theory. They are intimately related to symmetric functions, free probability, and Appell sequences through the raising operator R = x - d log(H(D))/dD for the Appell sequence inverse pair associated to the e.g.f.s H(t)e^(xt) (cf. A094587) and (1/H(t))e^(xt) with H(0)=1.

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, 2009.

F. Hirzebruch, Topological methods in algebraic geometry. Second, corrected printing of the third edition. Die Grundlehren der Mathematischen Wissenschaften, Band 131 Springer-Verlag, Berlin Heidelberg New York, 1978, p. 11 and 92.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..5762

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

H. Airault, Remarks on Faber polynomials, International Mathematical Forum, 3, no. 9, 2008, pages 449-456.

H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, pages 179-222.

H. Airault and Y. Neretin, On the action of the Virasoro algebra on the space of univalent functions, arXiv:0704.2149 [math.RT], 2007.

F. Ardila, Algebraic and geometric methods in enumerative combinatorics, arXiv:1409.2562 [math.CO], 2015.

A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, pages 49-70.

V. Chan, Topological K-theory of complex projective spaces, senior's thesis UC Davis (p. 11 on Chern characters of complex vector bundles), 2013.

T. Copeland, Lagrange a la Lah, 2011.

T. Copeland, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants, MathOverflow question, 2012.

T. Copeland, Addendum to Elliptic Lie Triad, 2015.

T. Copeland, Connection  between the Chebyshev polynomials and the Faber polynomials, MathOverflow question, 2015.

A. Dress and C. Siebeneicher, The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, lambda-rings, and the universal ring of Witt vectors, Adv. in Math., Vol. 78, Issue 1, Nov. 1989, pages 1-41.

D. Dugger, A Geometric Introduction to K-Theory, p. 288 (called Newton polynomials).

R. Friedrich and J. McKay, Formal groups, Witt vectors and free probablility, arXiv:1204.6522 [math.OA], 2012.

J. McKay and A. Sebbar, On replicable functions: an introduction, Frontiers in Number Theory, Physics, and Geometry II, pp. 373-386.

T. Takebe, Lee-Peng Teo, A. Zabrodin, Löwner equations and dispersionless hierarchies, arXiv:math/0605161 [math.CV], p. 24, 2006.

Lee-Peng Teo, Analytic functions and integrable hierarchies-characterization of tau functions, Letters in Mathematical Physics,  Vol. 64, Issue 1, Apr 2003, pp. 75-92 (also arXiv:hep-th/0305005, 2003).

Wikipedia, Newton identities.

FORMULA

-log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n.

-d(1 + b(1) x + b(2) x^2 + ...)/dx / (1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) x^(n-1).

F(n,b(1),..,b(n)) = -n*b(n) - Sum_{k=1..n-1} b(n-k)*F(k,b(1),..,b(k)).

Umbrally, with B(x) = 1 + b(1) x + b(2) x^2 + ..., B(x) = exp[log(1-F.x)] and 1/B(x) = exp[-log(1-F.x)], establishing a connection to the e.g.f. of A036039 and the symmetric polynomials.

The Stirling partition polynomials of the first kind St1(n,b1,b2,...,bn;-1) = IF(n,b1,b2,...,bn) (cf. the Copeland link Lagrange a la Lah, signed A036039, and p. 184 of Airault and Bouali), i.e., the cyclic partition polynomials for the symmetric groups, and the Faber polynomials form an inverse pair for isolating the indeterminates in their definition, for example, F(3,IF(1,b1),IF(2,b1,b2)/2!,IF(3,b1,b2,b3)/3!)= b3, with bk = b(k), and IF(3,F(1,b1),F(2,b1,b2),F(3,b1,b2,b3))/3!= b3.

The polynomials specialize to F(n,t,t,...) = (1-t)^n - 1.

See Newton Identities on Wikipedia on relation between the power sum symmetric polynomials and the complete homogeneous and elementary symmetric polynomials for an expression in multinomials for the coefficients of the Faber polynomials.

(n-1)! F(n,x[1],x[2]/2!,...,x[n]/n!) = - p_n(x[1],..,x[n]), where p_n are the cumulants of A127671 expressed in terms ot the moments x[n]. - Tom Copeland, Nov 17 2015

-(n-1)! F(n,B(1,x[1]),B(2,x[1],x[2])/2!,..,B(n,x[1],..,x[n])/n!) = x[n] provides an extraction of the indeterminates of the complete Bell partition polynomials B(n,x[1],...,x[n]) of A036040. Conversely, IF(n,-x[1],-x[2],-x[3]/2!,..,-x[n]/(n-1)!) = B(n,x[1],..,x[n]). - Tom Copeland, Nov 29 2015

For a square matrix M, determinant(I - x M) = exp[-sum(k>0, trace(M^k) x^k / k)] = sum[n>0, P_n(-trace(M),-trace(M^2),..,-trace(M^n)) x^n/n! ] = 1 + sum(n>0, d[n] x^n), where P_n(x[1],..,x{n]) are the cycle index partition polynomials of A036039 and d[n] = P_n(-trace(M),-trace(M^2),..,-trace(M^n)) / n!. Umbrally, det(I - x M)= exp[log(1 - b. x)] = exp[P.(-b_1,..,-b_n)x] = 1 / (1-d.x), where b_k = tr(M^k). Then F(n,d[1],..,d[n]) = tr[M^n]. - Tom Copeland, Dec 04 2015

Given f(x) = -log(g(x)) = -log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n, action on u_n = F(n,b(1),...,b(n)) with A133932 gives the compositional inverse finv(x) of f(x), with F(1,b(1)) not equal to zero, and f(g(finv(x))) = f(e^(-x)). Note also that exp(f(x)) = 1 / g(x) = exp[Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n] implies relations among A036040, A133314, A036039, and the Faber polynomials. - Tom Copeland, Dec 16 2015

The Dress and Siebeneicher paper gives combinatorial interpretations and various relations that the Faber polynomials must satisfy for integral values of its arguments. E.g., Eqn. (1.2) p. 2 implies [2 * F(1,-1) + F(2,-1,b2) + F(4,-1,b2,b3,b4)]  mod(4) = 0. This equation implies that [F(n,b1,b2,..,bn)-(-b1)^n] mod(n) = 0 for n prime. - Tom Copeland, Feb 01 2016

With the elementary Schur polynomials S(n,a_1,a_2,..,a_n) = Lah(n,a_1,a_2,..,a_n) / n!, where Lah(n,..) are the refined Lah polynomials of A130561, F(n,S(1,a_1),S(2,a_1,a_2),..,S(n,a_1,..,a_n)) = -n * a_n since sum_{n > 0} a_n x^n = log[sum{n >= 0} S(n,a_1,..,a_n) x^n]. Conversely, S(n,-F(1,a_1),-F(2,a_1,a_2)/2,..,-F(n,a_1,..,a_n)/n) = a_n. - Tom Copeland, Sep 07 2016

See Corollary 3.1.3 on p. 38 of Ardila and Copeland's two MathOverflow links to relate the Faber polynomials, with arguments being the signed elementary symmetric polynomials, to the logarithm of determinants, traces of powers of an adjacency matrix, and number of walks on graphs. - Tom Copeland, Jan 02 2017

EXAMPLE

F(1,b1) = - b1

F(2,b1,b2) = -2 b2 + b1^2

F(3,b1,b2,b3) = -3 b3 + 3 b1 b2 - b1^3

F(4,b1,..) = -4 b4 + 4 b1 b3 + 2 b2^2  - 4 b1^2 b2 + b1^4

F(5,..) = -5 b5 + 5 b1 b4 + 5 b2 b3 - 5 b1^2 b3 - 5 b1 b2^2 + 5 b1^3 b2 - b1^5

------------------------------

IF(1,b1) = -b1

IF(2,b1,,b2) = -b2 + b1^2

IF(3,b1,b2,b3) = -2 b3 + 3 b1 b2 - b1^3

IF(4,b1,...) = -6 b4 + 8 b1 b3 + 3 b2^2  - 6 b1^2 b2 + b1^4

IF(5,..) = -24 b5 + 30 b1 b4 + 20 b2 b3 - 20 b1^2 b3 - 15 b1 b2^2 + 10 b1^3 b2 - b1^5

------------------------------

For 1/(1+x)^2 = 1- 2x + 3x^2 - 4x^3 + 5x^4 - ... , F(n,-2,3,-4,...) = (-1)^(n+1) 2.

------------------------------

F(n,x,2x,...,nx), F(n,-x,2x,-3x,...,(-1)^n n*x), and F(n,(2-x),1,0,0,...) are related to the Chebyshev polynomials through A127677 and A111125. See also A110162, A156308, A208513, A217476, and A220668.

------------------------------

For b1 = p, b2 = q, and all other indeterminates 0, see A113279 and A034807.

For b1 = -y, b2 = 1 and all other indeterminates 0, see A127672.

MATHEMATICA

F[0] = 1; F[1] = -b[1]; F[2] = b[1]^2 - 2 b[2]; F[n_] := F[n] = -b[1] F[n - 1] - Sum[b[n - k] F[k], {k, 1, n - 2}] - n b[n] // Expand;

row[n_] := (List @@ F[n]) /. b[_] -> 1 // Reverse;

Table[row[n], {n, 1, 8}] // Flatten // Rest (* Jean-François Alcover, Jun 12 2017 *)

CROSSREFS

Cf. A036039, A036040, A115131, A210258, A127677, A110162, A156308, A208513, A217476, A220668, A113279, A034807, A111125, A127672, A094587, A127671, A133314.

Cf. A130561.

Sequence in context: A054531 A207645 A115131 * A210258 A181108 A211782

Adjacent sequences:  A263913 A263914 A263915 * A263917 A263918 A263919

KEYWORD

sign,tabf,easy

AUTHOR

Tom Copeland, Oct 29 2015

EXTENSIONS

More terms from Jean-François Alcover, Jun 12 2017

STATUS

approved

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Last modified June 20 22:19 EDT 2018. Contains 305615 sequences. (Running on oeis4.)