A115457 -id:A115457 - cp's OEIS frontend

A152128 Duplicate of A115457.

Original entry on oeis.org

0, 6, 35, 694, 26089, 1862994, 253247715, 66799608630, 34698378752226, 35781375988234520, 73534241823793715433

Offset: 0

Jonathan Vos Post, Nov 25 2008

dead

This is from Hou and Mullen, p.29, Table 1: I_2(n), q = 2.

Xiang-dong Hou and Gary L. Mullen, Number of Irreducible Polynomials and Pairs of Relatively Prime Polynomials in Several Variables over Finite Fields, arXiv:0811.3986 [math.NT], 2008.

Cf. A115457, A152129, A152130, A152131.

Apparently a(n)=A115457(n), n>0. [R. J. Mathar, Dec 03 2008]

A115505 Number of monic irreducible polynomials of degree 2 in GF(2^n)[x1,x2,x3,x4,x5].

Original entry on oeis.org

2095135, 1466014571406, 1317624575992306876, 1289520874254978952587000, 1308542555074300717818898758640, 1350326852860866917910548456417128416, 1404771351088543190017398373020865731215296

Offset: 1

Max Alekseyev, Jan 16 2006

nonn

Max Alekseyev, Formula for the number of monic irreducible polynomials in a finite field
Max Alekseyev, PARI scripts for various problems

Cf. A115457-A115504.

A122743 Number of normalized polynomials of degree n in GF(2)[x,y].

Original entry on oeis.org

1, 6, 56, 960, 31744, 2064384, 266338304, 68451041280, 35115652612096, 35993612646875136, 73750947497819242496, 302157667927362455470080, 2475577847115856892504571904, 40562343327224770087344704323584, 1329187430965708569562959165777772544

Offset: 0

N. J. A. Sloane, Aug 13 2008

nonn

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
a(n) is the HANKEL transform of the large Schröder numbers A006318(n+2). - Emanuele Munarini, Sep 14 2017

			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.

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.

Arnaud Bodin, Number of irreducible polynomials in several variables over finite fields, arXiv:0706.0157 [math.AC], 2007; 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, arXiv:0912.3312 [math.AC], 2009-2013.

Cf. A115457, A203011.
Row sums of powers of two triangles A000079.
Equals A000225(n+1)*2^A000217(n).

[2^((n+1)*(n+2) div 2) - 2^(n*(n+1) div 2): n in [0..30]]; // Vincenzo Librandi, Oct 01 2015

seq(2^((n*(1+n))/2)*(2^(1+n)-1), n=0..14); # Peter Luschny, Sep 19 2017

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 *)

a(n) = 2^((n+1)*(n+2)/2) - 2^(n*(n+1)/2);
vector (100, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

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
From Emanuele Munarini, Sep 14 2017: (Start)
(2^(n+1)-1)*a(n+1) - 2^(n+1)*(2^(n+2)-1)*a(n) = 0.
a(n+1) - (2^(n+2)+1)*a(n) = 2^(binomial(n+1,2)).
a(n+2) - (5*2^(n+1)+1)*a(n+1) + 2^(n+1)*(2^(n+2)+1)*a(n) = 0. (End)

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
a(14) from Vincenzo Librandi, Oct 01 2015

A115490 Number of monic irreducible polynomials of degree 4 in GF(2^n)[x].

Original entry on oeis.org

3, 60, 1008, 16320, 261888, 4193280, 67104768, 1073725440, 17179803648, 274877644800, 4398045462528, 70368739983360, 1125899890065408, 18014398442373120, 288230375883276288, 4611686017353646080, 73786976290543239168, 1180591620700231434240, 18889465931409861378048

Offset: 1

Max Alekseyev, Jan 16 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..800
Max Alekseyev, Formula for the number of monic irreducible polynomials in a finite field
Max Alekseyev, PARI scripts for various problems
Steven T. Dougherty and Esengül Saltürk, The neighbor graph of binary self-orthogonal codes, Adv. Math. Comm. (2024). See p. 16.
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A115457-A115505.

[(16^n-4^n)/4: n in [1..20]]; // Vincenzo Librandi, Jul 25 2014

CoefficientList[Series[3/((16 x - 1) (4 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2014 *)

a(0)=0, a(1)=3; for n>1, a(n) = 20*a(n-1)-64*a(n-2). - T. D. Noe, Nov 30 2006
G.f.: 3*x / ( (16*x-1)*(4*x-1) ). - R. J. Mathar, Jul 23 2014
a(n) = (16^n-4^n)/4. - Vincenzo Librandi, Jul 25 2014
E.g.f.: exp(4*x)*(exp(12*x) - 1)/4. - Stefano Spezia, Aug 16 2024

A115478 Number of monic irreducible polynomials of degree 3 in GF(3)[x1,...,xn].

Original entry on oeis.org

8, 25520, 1742232560, 25015771681981520, 261673816513678364838549056, 5986257591281009894301357511191882167288, 898505149957215605206589914754802519619582611893609512640

Offset: 1

Max Alekseyev, Jan 16 2006

nonn

Max Alekseyev, Formula for the number of monic irreducible polynomials in a finite field
Max Alekseyev, PARI scripts for various problems

Cf. A115457-A115505.

A115489 Number of monic irreducible polynomials of degree 3 in GF(2^n)[x].

Original entry on oeis.org

2, 20, 168, 1360, 10912, 87360, 699008, 5592320, 44739072, 357913600, 2863310848, 22906490880, 183251935232, 1466015498240, 11728124018688, 93824992215040, 750599937851392, 6004799503073280, 48038396025110528

Offset: 1

Max Alekseyev, Jan 16 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Max Alekseyev, Formula for the number of monic irreducible polynomials in a finite field
Max Alekseyev, PARI scripts for various problems
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A115457-A115505.

[-(1/3)*2^n+(1/3)*8^n: n in [1..20]]; // Vincenzo Librandi, Jul 25 2014

LinearRecurrence[{10,-16},{2,20},30] (* Harvey P. Dale, Sep 25 2013 *)
CoefficientList[Series[2/((8 x - 1) (2 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 25 2014 *)

a(0)=0, a(1)=2, a(n)=10a(n-1)-16a(n-2). - T. D. Noe, Nov 30 2006

G.f.: 2*x / ( (8*x-1)*(2*x-1) ). - R. J. Mathar, Jul 23 2014

A115504 Number of monic irreducible polynomials of degree 1 in GF(2^n)[x1,x2,x3,x4,x5].

Original entry on oeis.org

62, 1364, 37448, 1118480, 34636832, 1090785344, 34630287488, 1103823438080, 35253226045952, 1127000493261824, 36046397799139328, 1153203048319815680, 36897992296869404672, 1180663682709764194304

Offset: 1

Max Alekseyev, Jan 16 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..670
Max Alekseyev, Formula for the number of monic irreducible polynomials in a finite field
Max Alekseyev, PARI scripts for various problems
Index entries for linear recurrences with constant coefficients, signature (62,-1240,9920,-31744,32768).

Cf. A115457-A115505.

[2^n+4^n+8^n+16^n+32^n: n in [1..20]]; // Vincenzo Librandi, Jul 25 2014

CoefficientList[Series[-2 (31 - 1240 x + 14880 x^2 - 63488 x^3 + 81920 x^4)/((4 x - 1) (2 x - 1) (8 x - 1) (16 x - 1) (32 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 25 2014 *)
LinearRecurrence[{62,-1240,9920,-31744,32768},{62,1364,37448,1118480,34636832},20] (* Harvey P. Dale, Oct 07 2019 *)

a(n) = A034665(n) - 1, or a(n) = 2^n + 4^n + 8^n + 16^n + 32^n. - Chris Boyd, Apr 26 2014

G.f.: -2*x*( 31-1240*x+14880*x^2-63488*x^3+81920*x^4 ) / ( (4*x-1)*(2*x-1)*(8*x-1)*(16*x-1)*(32*x-1) ). - R. J. Mathar, Jul 23 2014

A115458 Number of monic irreducible polynomials of degree n in GF(2)[x,y,z].

Original entry on oeis.org

1, 14, 903, 1034350, 34343703541, 72057077817486762, 19342812032942216095260047, 1329227995494773618659262698956301950, 46768052394568954962565705269783921192917309706498

Offset: 0

Max Alekseyev, Jan 16 2006

nonn

Max Alekseyev, Formula for the number of monic irreducible polynomials in a finite field
Max Alekseyev, PARI scripts for various problems
J. von zur Gathen and K. Ziegler, Survey on counting special types of polynomials, arXiv preprint arXiv:1407.2970 [math.AC], 2014.
Konstantin Ziegler, Counting Classes of Special Polynomials, Doctoral Dissertation, University of Bonn, June 2014.

Cf. A115457-A115505.

A115461 Number of monic irreducible polynomials of degree n in GF(3)[x,y].

Original entry on oeis.org

1, 12, 273, 25520, 6778629, 5132148528, 11368775698280, 74897449398451680, 1476178370884382958936, 87205387550224830516286800, 15450442981642705273095610563240, 8211400746584626963688710296061894800

Offset: 0

Max Alekseyev, Jan 16 2006

nonn

Max Alekseyev, Formula for the number of monic irreducible polynomials in a finite field
Max Alekseyev, PARI scripts for various problems
J. von zur Gathen and K. Ziegler, Survey on counting special types of polynomials, arXiv preprint arXiv:1407.2970 [math.AC], 2014.
Konstantin Ziegler, Counting Classes of Special Polynomials, Doctoral Dissertation, University of Bonn, June 2014.

Cf. A115457-A115505.

A115465 Number of monic irreducible polynomials of degree n in GF(5)[x,y].

Original entry on oeis.org

1, 30, 3410, 2330240, 7549603600, 118965950703744, 9309505329218297280, 3637689729211851543816960, 7105314552536912564123328420000, 69388718760088702173445263653542192000

Offset: 0

Max Alekseyev, Jan 16 2006

nonn

Max Alekseyev, Formula for the number of monic irreducible polynomials in a finite field
Max Alekseyev, PARI scripts for various problems
J. von zur Gathen and K. Ziegler, Survey on counting special types of polynomials, arXiv preprint arXiv:1407.2970 [math.AC], 2014.
Xiang-dong Hou and Gary L. Mullen, Number of Irreducible Polynomials and Pairs of Relatively Prime Polynomials in Several Variables over Finite Fields, arXiv:0811.3986 [math.NT], 2008.
Konstantin Ziegler, Counting Classes of Special Polynomials, Doctoral Dissertation, University of Bonn, June 2014.

Cf. A115457-A115505.

cp's OEIS Frontend

A152128 Duplicate of A115457.

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A115505 Number of monic irreducible polynomials of degree 2 in GF(2^n)[x1,x2,x3,x4,x5].

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A122743 Number of normalized polynomials of degree n in GF(2)[x,y].

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A115490 Number of monic irreducible polynomials of degree 4 in GF(2^n)[x].

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A115478 Number of monic irreducible polynomials of degree 3 in GF(3)[x1,...,xn].

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A115489 Number of monic irreducible polynomials of degree 3 in GF(2^n)[x].

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A115504 Number of monic irreducible polynomials of degree 1 in GF(2^n)[x1,x2,x3,x4,x5].

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A115458 Number of monic irreducible polynomials of degree n in GF(2)[x,y,z].

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A115461 Number of monic irreducible polynomials of degree n in GF(3)[x,y].

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A115465 Number of monic irreducible polynomials of degree n in GF(5)[x,y].