The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A008670

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A008670

Molien series for Weyl group F_4.
(history; published version)

#56 by Charles R Greathouse IV at Thu Sep 08 08:44:36 EDT 2022

PROG

(MAGMAMagma) MolienSeries(CoxeterGroup("F4")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

(MAGMAMagma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 08 2019

Discussion

Thu Sep 08

08:44

OEIS Server: https://oeis.org/edit/global/2944

#55 by Bruno Berselli at Mon Sep 09 04:32:44 EDT 2019

STATUS

reviewed

approved

#54 by Michel Marcus at Mon Sep 09 01:34:52 EDT 2019

STATUS

proposed

reviewed

#53 by G. C. Greubel at Sun Sep 08 16:00:59 EDT 2019

STATUS

editing

proposed

#52 by G. C. Greubel at Sun Sep 08 16:00:11 EDT 2019

LINKS

<a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, 0, -1, 1, -2, 1, -1, 0, 1, 0, 1, -1).

PROG

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 08 2019

(PARI) my(x='x+O('x^70)); Vec(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))) \\ G. C. Greubel, Sep 08 2019

(Sage)

def A008670_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))).list()

A008670_list(70) # G. C. Greubel, Sep 08 2019

CROSSREFS

Cf. A010875, A002411, A006002, A010875, A011934, A027480, A055232, A182260.

STATUS

approved

editing

#51 by N. J. A. Sloane at Mon Sep 10 04:44:21 EDT 2018

STATUS

editing

approved

#50 by N. J. A. Sloane at Mon Sep 10 04:44:18 EDT 2018

FORMULA

G.f.: 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)). [corrected Corrected by Ralf Stephan, Apr 29 2014]

a(n) ~ (1/432)*n^3. - Ralf Stephan, Apr 29 2014

a(n) = (120*floor(n/6)^3 + 60*(m+7)*floor(n/6)^2 + 2*(m^5-15*m^4+75*m^3-135*m^2+134*m+240)*floor(n/6) + 3*(m^5-15*m^4+75*m^3-135*m^2+84*m+70) + (m^5-15*m^4+75*m^3-135*m^2+44*m+30)*(-1)^floor(n/6))/240 where m = (n mod 6). - Luce ETIENNE, Aug 14 2018

STATUS

proposed

editing

#49 by Michel Marcus at Tue Aug 14 05:23:47 EDT 2018

STATUS

editing

proposed

Discussion

Wed Aug 15

01:20

Michel Marcus: I don't see why these xrefs (can you say here)

06:21

Luce ETIENNE: 𝑎(6𝑛)  = 1 5 16 36 69 117 184 272 385 525 696  see....    A055232(n) 
(𝑎(6𝑛+1) = 1 6 18 40 75 126 196 288 405 550 726 936 1183 1470.... see A002411(n+1) 
𝑎(6𝑛+2)  = 1 7 20 44 81 135 208 304 425 575 756 972 1225 1519 .... see  A011934(n+1) 
𝑎(6𝑛+3)  =  2 9 24 50 90 147 224 324 450 605 792 1014 1274 1575....  see   A006002(n+1)
 a(6n+4)  =  3 11 28 56 99 159 240 344 475 635 828 1056 1323 1631 see.... A182260(n+2) 
a(6n+5) =  3 12 30 60 105 168 252 360 495 660 858 1092 1365 1680 see.... A027480" (𝑛+1))))

#48 by Michel Marcus at Tue Aug 14 05:23:39 EDT 2018

LINKS

G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.

CROSSREFS

A010875;

Cf. A010875, A002411, A006002, A011934, A027480, A055232, A182260.

STATUS

proposed

editing

#47 by Luce ETIENNE at Tue Aug 14 05:20:24 EDT 2018

STATUS

editing

proposed