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

0,2

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

Arkadij L. Onishchik and Ernest B. 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

Joseph M. Landsberg and Laurent Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=4]

Samuel Lewis and Pavel Shlykov, Nakajima quiver varieties in dimension four, arXiv:2510.15160 [math.AG], 2025. See p. 30.

Formula

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

Sum_{n>=0} 1/a(n) = 886985599500*zeta(3) - 294239058325973501/275968. - Amiram Eldar, Jun 16 2026

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 *)

Prog

(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

Crossrefs

Cf. A121736.

Sequence in context: A025597 A034202 A160290 * A319310 A022081 A017772

Adjacent sequences: A030646 A030647 A030648 * A030650 A030651 A030652

Keyword

nonn,easy,changed

Author

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

Extensions

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

Status

approved