A178779 -id:A178779 - cp's OEIS frontend

A178795 Expansion of the polynomial (x^15-1)(x^12-1)(x^10-1)(x^9-1)(x^7-1)(x^6-1)(x^4-1)*(x-1) in increasing powers of x.

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 1, -2, 3, -1, -1, 3, -3, 1, 1, -4, 3, -1, -3, 4, -4, 1, 3, -5, 4, 0, -3, 6, -3, 0, 4, -5, 3, 1, -4, 4, -3, -1, 3, -4, 1, 1, -3, 3, -1, -1, 3, -2, 1, 1, -1, 1, 0, -1, 1, -1, 0, 0, -1, 1

Offset: 0

N. J. A. Sloane, Dec 26 2010

q^120*(q^30-1)*(q^24-1)*(q^20-1)*(q^18-1)*(q^14-1)*(q^12-1)*(q^8-1)*(q^2-1) is the order of the simple group E_8(q), if q is a prime power.

If f(x) is the x-polynomial and g(q) the q-polynomial, then g(q) = q^120*f(q^2). - Jean-François Alcover, Aug 25 2022

			With p=2 one gets the order of E_8(2): 337804753143634806261388190614085595079991692242467651576160959909068800000. - _Jean-François Alcover_, Aug 25 2022

R. L. Griess, Jr., Twelve Sporadic Groups, Springer, 1998; see p. 169.

Wikipedia, Lie group E8

Cf. A009968, A178779, A178780, A178781.

Vec((x^15-1)*(x^12-1)*(x^10-1)*(x^9-1)*(x^7-1)*(x^6-1)*(x^4-1)*(x-1)) \\ Michel Marcus, Aug 25 2022

A008913 Order of simple Chevalley group F_4(q), q = prime power.

Original entry on oeis.org

3311126603366400, 5734420792816671844761600, 19009825523840945451297669120000, 2131486317725501953125000000000000000, 86325573304608766361629193317905069834240000

Offset: 1

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.
R. L. Griess, Jr., Twelve Sporadic Groups, Springer, 1998; see p. 169.

Cf. A178779.

Maple
```
q^24*(q^2-1)*(q^6-1)*(q^8-1)*(q^12-1);
```

Table[q^24*(q^2-1)*(q^6-1)*(q^8-1)*(q^12-1), {q, Select[Range[7], PrimePowerQ]}] (* Jean-François Alcover, Aug 24 2022 *)

A178780 Expansion of the polynomial x^36(x^12-1)(x^9-1)(x^8-1)(x^6-1)(x^5-1)(x^2-1) in increasing powers of x.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, -1, -1, 1, 0, -1, 1, 2, -1, 0, 3, 0, -2, 1, 1, -3, -1, 2, -1, -3, 1, 1, -2, 0, 3, 0, -1, 2, 1, -1, 0, 1, -1, -1, 0, 0, -1, 0, 1

Offset: 0

N. J. A. Sloane, Dec 26 2010

q^36*(q^12-1)*(q^9-1)*(q^8-1)*(q^6-1)*(q^5-1)*(q^2-1) is the order of the simple group E_6(q), if q is a prime power.

R. L. Griess, Jr., Twelve Sporadic Groups, Springer, 1998; see p. 169.

Cf. A008871, A178779, A178781, A178795.

cp's OEIS Frontend

A178795 Expansion of the polynomial (x^15-1)(x^12-1)(x^10-1)(x^9-1)(x^7-1)(x^6-1)(x^4-1)*(x-1) in increasing powers of x.

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A008913 Order of simple Chevalley group F_4(q), q = prime power.

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A178780 Expansion of the polynomial x^36(x^12-1)(x^9-1)(x^8-1)(x^6-1)(x^5-1)(x^2-1) in increasing powers of x.

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cp's OEIS Frontend

A178795 Expansion of the polynomial (x^15-1)*(x^12-1)*(x^10-1)*(x^9-1)*(x^7-1)*(x^6-1)*(x^4-1)*(x-1) in increasing powers of x.

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PARI

A008913 Order of simple Chevalley group F_4(q), q = prime power.

Views

Author

Keywords

References

Crossrefs

Programs

Maple

Mathematica

A178780 Expansion of the polynomial x^36*(x^12-1)*(x^9-1)*(x^8-1)*(x^6-1)*(x^5-1)*(x^2-1) in increasing powers of x.

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Author

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A178795 Expansion of the polynomial (x^15-1)(x^12-1)(x^10-1)(x^9-1)(x^7-1)(x^6-1)(x^4-1)*(x-1) in increasing powers of x.

A178780 Expansion of the polynomial x^36(x^12-1)(x^9-1)(x^8-1)(x^6-1)(x^5-1)(x^2-1) in increasing powers of x.