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

0,2

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).

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A133355.

Sequence in context: A202289 A010943 A022622 * A223278 A278091 A042410

Adjacent sequences: A030645 A030646 A030647 * A030649 A030650 A030651

Keyword

nonn,easy

Author

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

Extensions

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

Status

approved