A079293 Degree of the numerator of Fn(z), the Poincaré series (also Hilbert, Molien series) for C(Vn)^G, where G = SL(2,C) and Vd is the space for binary forms of degree d.

0, 0, 0, 18, 15, 48, 18, 66, 48, 102, 52, 146, 83, 192, 102, 252, 136, 320, 168, 396, 210, 480, 250, 572, 300, 672, 348, 780, 406, 896, 462, 1020, 528, 1152, 592, 1292, 666, 1440, 738

Offset

2,4

Comments

Fn(z) is a rational function of degree -(n+1). Recently Brouwer, Cohen, and later Sally Jr. calculated Fn(z) for all n <= 18 and n = 20, 22, 24. It is rumored that Littelmann, Procesi, Laurent have calculated Fn(z) for many other values of n.

This sequence is somewhat badly defined. The values 18, 15, 48, 18 are not the degrees of the numerator of this rational function in lowest terms, but rather are degrees of a "representative" form. But there may be several representative forms with different degrees. - Andries E. Brouwer, Jan 15 2009

References

Jean-Michel Kantor, Où en sont les mathématiques?, SMF, Vuibert, Chapitre 5, paragraphe 6, "Invariants des formes binaires : la formule de Cayley-Sylvester", pp. 73-74.

Links

Table of n, a(n) for n=2..40.

Andries Brouwer, Poincaré Series.

A. E. Brouwer and A. M. Cohen, The Poincaré series of the polynomials invariant under SU2 in its irreducible representation of degree <=17, report ZW134, Math. Centr. Amsterdam, Dec. 1979.

J. J. Sylvester, Proof of the hitherto undemonstrated fundamental theorem of invariants, Phil. Mag. 30(5) (1878), 178-188.

J. J. Sylvester, Proof of the hitherto undemonstrated fundamental theorem of invariants, Phil. Mag. 30(5) (1878), 178-188.

Example

F8(z) = (1 + z^8 + z^9 + z^10 + z^18)/Product_{i = 2..7} (1-z^i), hence a(8) = 18. (See A097851. - N. J. A. Sloane, Oct 02 2021)

Crossrefs

Cf. A097851.

Sequence in context: A077668 A298723 A299557 * A070646 A375676 A094381

Adjacent sequences: A079290 A079291 A079292 * A079294 A079295 A079296

Keyword

more,nonn

Author

Benoit Cloitre, Feb 08 2003

Status

approved