A012249 Volume of a certain rational polytope whose points with given denominator count certain sets of Standard Tableaux.

1, 2, 5, 24, 154, 1280, 13005, 156800, 2189726, 34793472, 620169186, 12259602432, 266267950740, 6304157663232, 161624247752253, 4461403146190848, 131936409635518774, 4161949856324648960, 139508340802911502422, 4952126960969786064896

Offset

1,2

Comments

It should be noticed that Richard Stanley's formula (cf. A012250) gives a(9) = 2189726 instead of 2189725 as given in Verma (1997). - Jean-François Alcover, Nov 28 2013

Links

G. C. Greubel, Table of n, a(n) for n = 1..350

MathOverflow, Access to a preprint by D. N. Verma, Feb 2013.

D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - N. J. A. Sloane, Oct 03 2021]

Formula

a(n) ~ 3^(3/2) * 2^(n+1) * n^(n-2) / exp(n). - Vaclav Kotesovec, Oct 07 2021

a(n) = 2^(n-2)*Sum_{j=0..ceiling(n/2)} (-1)^(j+1)*(n/2-j+1)^(n-1) * binomial(n+2, j) (based on Richard Stanley's formula in A012250). - Jean-François Alcover, Nov 25 2013

Maple

A012249 := proc(n)
     add( (-1)^(j+1)*(n/2-j+1)^(n-1)*binomial(n+2, j), j=0..ceil(n/2)) ;
     %*2^(n-2) ;
end proc:
seq(A012249(n), n=1..20) ; # R. J. Mathar, Oct 07 2021

Mathematica

a[n_] := 2^(n-2)*Sum[(-1)^(j+1)*(n/2-j+1)^(n-1)*Binomial[n+2, j], {j, 0, Ceiling[n/2]}]; Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Nov 25 2013, after Richard Stanley's formula in A012250. *)

Prog

(Magma)
A012249:= func< n | 2^(n-2)*(&+[(-1)^(j+1)*Binomial(n+2, j)*(n/2-j+1)^(n-1) : j in [0..1+Floor(n/2)]] ) >;
[A012249(n): n in [1..30]]; // G. C. Greubel, Feb 28 2024
(SageMath)
def A012249(n): return 2^(n-2)*sum( (-1)^(j+1)*binomial(n+2, j)*(n/2-j+1)^(n-1) for j in range(n//2+2))
[A012249(n) for n in range(1, 31)] # G. C. Greubel, Feb 28 2024

Crossrefs

Cf. A012250.

Row sums of A348211.

Sequence in context: A054131 A086635 A020124 * A303108 A331035 A052808

Adjacent sequences: A012246 A012247 A012248 * A012250 A012251 A012252

Keyword

nonn,more

Author

D n Verma

Extensions

Corrected and extended by R. J. Mathar, Oct 07 2021

Edited by N. J. A. Sloane, Oct 07 2021

Status

approved