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
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
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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
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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)];
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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