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A122743 Number of normalized polynomials of degree n in GF(2)[x,y]. 6
1, 6, 56, 960, 31744, 2064384, 266338304, 68451041280, 35115652612096, 35993612646875136, 73750947497819242496, 302157667927362455470080, 2475577847115856892504571904, 40562343327224770087344704323584 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = n-th elementary symmetric function in n+1 variables evaluated at {2,4,8,16,...,2^(n+1)}; see Mathematica program.

a(n) is the number of simple labeled graphs on {1,2,...,n+2} such that the vertex 1 is not isolated. - Geoffrey Critzer, Sep 12 2013

REFERENCES

Arnaud Bodin, Number of irreducible polynomials in several variables over finite fields, http://arxiv.org/abs/0706.0157, Amer. Math. Monthly, 115 (2008), 653-660.

Joachim von zur Gathen, Alfredo Viola, and Konstantin Ziegler, Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields, in: A. López-Ortiz (Ed.), LATIN 2010: Theoretical Informatics, Proceedings of the 9th Latin American Symposium, Oaxaca, Mexico, April 19-23, 2010, in: Lecture Notes in Comput. Sci., vol. 6034, Springer, Berlin, Heidelberg, 2010, pp. 243-254 (Extended Abstract). Final version to appear in SIAM J. Discrete Math.

LINKS

Table of n, a(n) for n=0..13.

FORMULA

a(n) = 2^((n+1)(n+2)/2) - 2^(n(n+1)/2). [Paul D. Hanna, Apr 08 2009]

E.g.f.: d(G(2x)-G(x))/dx where G(x) is the e.g.f. for A006125. - Geoffrey Critzer, Sep 12 2013

EXAMPLE

Let esp abbreviate "elementary symmetric polynomial".  Then

0th esp of {2} is 1.

1st esp of {2,4} is 2+4 = 6.

2nd esp of {2,4,8} is 2*4 + 2*8 + 4*8 = 56.

MATHEMATICA

f[k_] := 2^k; t[n_] := Table[f[k], {k, 1, n}]

a[n_] := SymmetricPolynomial[n - 1, t[n]]

Table[a[n], {n, 1, 16}] (* A122743 *)

(* Clark Kimberling, Dec 29 2011 *)

CROSSREFS

Cf. A115457, A203011.

Row sums of powers of two triangle A000079.

Equals A000225(n+1)*2^A000217(n).

Sequence in context: A093197 A052317 A185524 * A241031 A137032 A053421

Adjacent sequences:  A122740 A122741 A122742 * A122744 A122745 A122746

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 13 2008

EXTENSIONS

Edited, terms and links added by Johannes W. Meijer, Oct 10 2010

Comments corrected, reference added, and example edited by Konstantin Ziegler, Dec 04 2012

STATUS

approved

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Last modified November 27 15:49 EST 2014. Contains 250225 sequences.