A133238 Dimensions of certain Lie algebra (see reference for precise definition).

1, 52, 1053, 12376, 100776, 627912, 3187041, 13748020, 51949755, 175847880, 542393670, 1544927904, 4107092288, 10278624864, 24388573014, 55188666312, 119696471453, 249869263644, 503865726155, 984563860280, 1869304764600, 3456658569000, 6238533257775

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Joseph M. Landsberg and Laurent Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.1, case a=1]

Formula

Empirical g.f.: (x^8+36*x^7+341*x^6+1208*x^5+1820*x^4+1208*x^3+341*x^2+36*x+1) / (x-1)^16. - Colin Barker, Jul 27 2013

Sum_{n>=0} 1/a(n) = 12936000*zeta(3) - 2018508800*log(2)/9 + 22665157559/162. - Amiram Eldar, Jun 16 2026

Maple

b:=binomial; t71:= proc(a, k) ((3*a+2*k+5)/(3*a+5)) * b(k+2*a+3, k)*b(k+5*a/2+3, k)*b(k+3*a+4, k)/(b(k+a/2+1, k)*b(k+a+1, k)); end; [seq(t71(1, k), k=0..30)];

Mathematica

t71[a_, k_] := (3a+2k+5) / (3a+5) Binomial[k+2a+3, k] Binomial[k+5/2a+3, k] Binomial[k+3a+4, k] / (Binomial[k+a/2+1, k] Binomial[k+a+1, k]);
Array[t71[1, #]&, 30, 0] (* Paolo Xausa, Jan 11 2024 *)

Crossrefs

The cases a = -4/3, -1, -2/3, 0, 1, 2, 4, 6, 8 of Th. 7.1 of Landsberg and Manivel give sequences A005408, A000578, A085462, A107942, A133238 (this entry), A133239, A133240, A133241 and A030650 respectively. See also triangle in A128894.

Sequence in context: A083936 A273655 A160288 * A331127 A216939 A130000

Adjacent sequences: A133235 A133236 A133237 * A133239 A133240 A133241

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 15 2007

Status

approved