A030649 Dimensions of multiples of minimal representation of complex Lie algebra E7.

1, 56, 1463, 24320, 293930, 2785552, 21737254, 144538624, 839848450, 4347450800, 20355385710, 87265194240, 345992859975, 1279301331000, 4442249264625, 14573017267200, 45398364338250, 134897996890800, 383822534859750, 1049290591104000, 2764459117589400

Offset: 0

Author

Paolo Dominici (pl.dm(AT)libero.it)

Keywords

• nonn

Comments

From Alexander R. Povolotsky, Nov 19 2007: (Start)

After adjustment for the fact that a(n) is indexed from 0 while A121736 is indexed from 1, it appears that in many cases (with some exceptions) (a(n) - A121736(n+1))/133 (where A121736(3) = 133) yields integral values:

(1 - 1)/133 = 0

(56 - 56)/133 = 0

(1463 - 133) / 133 = 10

(24320 - 912) / 133 = 176

(293930 - 1463) / 133 = 2199

(2785552 - 1539) / 133 = 146527/7

(21737254 - 6480) / 133 = 21730774/133

(144538624 - 7371) / 133 = 144531253/133

(839848450 - 8645) / 133 = 6314585

(4347450800 - 24320) / 133 = 228811920/7

(20355385710 - 27664) / 133 = 153047805

(87265194240 - 40755) / 133 = 656128974

(345992859975 - 51072) / 133 = 2601449691

(1279301331000 - 86184) / 133 = 9618806352

(4442249264625 - 150822) / 133 = 233802584937/7

(14573017267200 - 152152)/133 = 109571557256

(45398364338250 - 238602)/133 = 341341083456

(134897996890800 - 253935)/133 = 1014270651405

(383822534565820 - 293930)/133 = 2885883718540

(1049290591104000 - 320112)/133 = 1049290590783888/133

...

Note that 133 is also the dimension of the Lie algebra E_7. (End)

References

• Onishchik and Vinberg, Seminar on Lie Groups and Algebraic Groups, Springer Verlag 1990, see Table 5.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=4]

Crossrefs

Cf. A121736.

Programs

Maple

b:=binomial; t72b:= proc(a,k) ((a+k+1)/(a+1)) * b(k+2*a+1,k)*b(k+3*a/2+1,k)/(b(k+a/2,k)); end; [seq(t72b(8,k),k=0..28)];

Mathematica

Table[(1/10950439500)*(n + 9)*Binomial[n + 17, 4]*Binomial[n + 4, 4]* Binomial[n + 13, 9]^2, {n,0,50}] (* G. C. Greubel, Feb 19 2017 *)

PARI

for(n=0,25, print1((1/10950439500)*(n+9)*binomial(n+17, 4)*binomial(n+4, 4)*binomial(n+13, 9)^2, ", ")) \\ G. C. Greubel, Feb 19 2017

Formula

a(n) = (1/10950439500)*(n+9)*binomial(n+17, 4)*binomial(n+4, 4)*binomial(n+13, 9)^2.

Extensions

Entry revised by N. J. A. Sloane, Oct 20 2007