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A012249 Volume of a certain rational polytope whose points with given denominator count certain sets of Standard Tableaux. 8
1, 2, 5, 24, 154, 1280, 13005, 156800, 2189726, 34793472, 620169186, 12259602432, 266267950740, 6304157663232, 161624247752253, 4461403146190848, 131936409635518774, 4161949856324648960, 139508340802911502422, 4952126960969786064896 (list; graph; refs; listen; history; text; internal format)
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
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D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished.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
KEYWORD
nonn,more,changed
AUTHOR
EXTENSIONS
Corrected and extended by R. J. Mathar, Oct 07 2021
Edited by N. J. A. Sloane, Oct 07 2021
STATUS
approved

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Last modified February 29 13:31 EST 2024. Contains 370425 sequences. (Running on oeis4.)