A030648 Dimensions of multiples of minimal representation of complex Lie algebra E6.

1, 27, 351, 3003, 19305, 100386, 442442, 1706562, 5895396, 18559580, 53965548, 146477916, 374332452, 907036326, 2096092350, 4642456390, 9895762305, 20373628275, 40639459575, 78751105875, 148599912825, 273612537900, 492502568220, 868056366060, 1500344336400

Offset: 0

Author

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

Keywords

• nonn
• easy

References

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

Links

• T. D. Noe, 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. 7.3, case a=8]
• Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Crossrefs

Cf. A133355.

Programs

Magma

A030648:= func< n | Binomial(n+8,8)*Binomial(n+11,8)/165 >;
[A030648(n): n in [0..30]]; // G. C. Greubel, Feb 09 2025

Maple

b:=binomial; t73:= proc(a,k) ((2*k+a)*(k+a)/(a^2)) * b(k+a-1,k)*b(k+3*a/2-1,k)/(b(k+a/2,k)); end; [seq(t73(8,k),k=0..40)];

Mathematica

Table[(Binomial[n+11,3]Binomial[n+3,3]Binomial[n+8,5]^2)/517440,{n,0,30}]  (* Harvey P. Dale, May 01 2011 *)

SageMath

def A030648(n): return binomial(n+8,8)*binomial(n+11,8)//165
print([A030648(n) for n in range(31)]) # G. C. Greubel, Feb 09 2025

Formula

a(n) = (1/517440)*binomial(n+11, 3)*binomial(n+3, 3)*binomial(n+8, 5)^2.

From G. C. Greubel, Feb 09 2025: (Start)

a(n) = (1/165)*binomial(n+8,8)*binomial(n+11,8).

G.f.: (1 + 10*x + 28*x^2 + 28*x^3 + 10*x^4 + x^5)/(1-x)^17. (End)

Extensions

Edited by N. J. A. Sloane, Oct 20 2007