A030647 Dimension of multiples of minimal representation of complex Lie algebra F4.

26, 324, 2652, 16302, 81081, 342056, 1264120, 4188834, 12664184, 35405968, 92512368, 227854536, 532703874, 1189056024, 2546364040, 5253305915, 10477865970, 20265831300, 38111646300, 69848806950, 125012625075, 218890030320, 375553797840, 632287970340, 1045954337376

Offset

1,1

References

Arkadij L. Onishchik and Ernest B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag Berlin Heidelberg, 1990. See table 5.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

a(n) = (1/1617000)*(2*n+11)*binomial(n+10, 3)*binomial(n+3, 3)*binomial(n+7, 4)^2.

G.f.: (x+1)*(x^4+9*x^3+19*x^2+9*x+1)/(x-1)^16. - Vaclav Kotesovec, Oct 21 2012

From Amiram Eldar, Oct 27 2025: (Start)

Sum_{n>=1} 1/a(n) = 26863174750/567 - 1845493760*log(2)/27.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2695000*Pi^2/3 - 461373440*Pi/27 + 50822526547/1134. (End)

Mathematica

Rest[CoefficientList[Series[(x+1)*(x^4+9*x^3+19*x^2+9*x+1)/(x-1)^16, {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 21 2012 *)

Prog

(PARI) my(x='x+O('x^66)); Vec(-1 + (x+1)*(x^4+9*x^3+19*x^2+9*x+1)/(x-1)^16) \\ Joerg Arndt, May 04 2013

Crossrefs

Sequence in context: A015800 A276285 A232064 * A202292 A010942 A022621

Adjacent sequences: A030644 A030645 A030646 * A030648 A030649 A030650

Keyword

nonn,easy

Author

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

Status

approved