A133317 Dimensions of certain Lie algebra (see reference for precise definition).

1, 35, 405, 2695, 12740, 47628, 149940, 413820, 1029105, 2351635, 5010005, 10061415, 19211920, 35119280, 61799760, 105163632, 173707785, 279397755, 438775645, 674334815, 1016206884, 1504211500, 2190324500, 3141625500, 4443791625, 6205210011, 8561787885

Offset

0,2

Comments

This is the case P(5,n) of the family of sequences defined in A132458. - Ottavio D'Antona (dantona(AT)dico.unimi.it), Oct 31 2007

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. 7.2(i), case a=2]

Formula

Empirical g.f.: (x+1)*(x^4+24*x^3+76*x^2+24*x+1) / (x-1)^10. - Colin Barker, Jul 27 2013

From Amiram Eldar, Jun 16 2026: (Start)

Sum_{n>=0} 1/a(n) = 81920*log(2)/9 - 170320/27.

Sum_{n>=0} (-1)^n/a(n) = 20480*Pi/9 - 1040*Pi^2/9 - 162200/27. (End)

Maple

b:=binomial; t72a:= proc(a, k) ((2*a+2*k+1)/(2*a+1)) * b(k+3*a/2-1, k)*b(k+3*a/2+1, k)*b(k+2*a, k)/(b(k+a/2-1, k)*b(k+a/2+1, k)); end; [seq(t72a(2, k), k=0..40)];

Mathematica

t72a[a_, k_] := (2k+2a+1) / (2a+1) Binomial[k+3/2a-1, k] Binomial[k+3/2a+1, k] Binomial[k+2a, k] / (Binomial[k+a/2-1, k] Binomial[k+a/2+1, k]);
Array[t72a[2, #]&, 30, 0] (* Paolo Xausa, Jan 10 2024 *)

Crossrefs

Cf. A132458.

Sequence in context: A380581 A238539 A075664 * A322879 A105947 A183846

Adjacent sequences: A133314 A133315 A133316 * A133318 A133319 A133320

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 19 2007

Status

approved