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, 185585825504872433198636

Offset: 1

Author

D n Verma

Keywords

• nonn

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• D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished.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]

Crossrefs

Cf. A012250.

Row sums of A348211.

Programs

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

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. *)

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

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

Extensions

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

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