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

%I A012249 #62 Jul 21 2025 14:38:11
%S A012249 1,2,5,24,154,1280,13005,156800,2189726,34793472,620169186,
%T A012249 12259602432,266267950740,6304157663232,161624247752253,
%U A012249 4461403146190848,131936409635518774,4161949856324648960,139508340802911502422,4952126960969786064896,185585825504872433198636
%N A012249 Volume of a certain rational polytope whose points with given denominator count certain sets of Standard Tableaux.
%C A012249 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
%H A012249 G. C. Greubel, <a href="/A012249/b012249.txt">Table of n, a(n) for n = 1..350</a>
%H A012249 MathOverflow, <a href="https://mathoverflow.net/questions/123179/access-to-a-preprint-by-d-n-verma">Access to a preprint by D. N. Verma</a>, Feb 2013.
%H A012249 D.-N. Verma, <a href="/A012249/a012249.pdf">Towards Classifying Finite Point-Set Configurations</a>, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 03 2021]
%F A012249 a(n) ~ 3^(3/2) * 2^(n+1) * n^(n-2) / exp(n). - _Vaclav Kotesovec_, Oct 07 2021
%F A012249 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
%p A012249 A012249 := proc(n)
%p A012249      add( (-1)^(j+1)*(n/2-j+1)^(n-1)*binomial(n+2,j),j=0..ceil(n/2)) ;
%p A012249      %*2^(n-2) ;
%p A012249 end proc:
%p A012249 seq(A012249(n),n=1..20) ; # _R. J. Mathar_, Oct 07 2021
%t A012249 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. *)
%o A012249 (Magma)
%o A012249 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)]] ) >;
%o A012249 [A012249(n): n in [1..30]]; // _G. C. Greubel_, Feb 28 2024
%o A012249 (SageMath)
%o A012249 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))
%o A012249 [A012249(n) for n in range(1,31)] # _G. C. Greubel_, Feb 28 2024
%Y A012249 Cf. A012250.
%Y A012249 Row sums of A348211.
%K A012249 nonn
%O A012249 1,2
%A A012249 _D n Verma_
%E A012249 Corrected and extended by _R. J. Mathar_, Oct 07 2021
%E A012249 Edited by _N. J. A. Sloane_, Oct 07 2021