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%I #51 Feb 28 2024 02:23:33
%S 1,3,40,1225,67956,5986134,769550496,136151219061,31753157473180,
%T 9445432588519642,3491687484842443536,1570713950508131878618,
%U 845034544811095556274280,535857105694970626486925100,395590680969537758258609408640,336386798400777928783348084420365
%N a(n) = A012249(2*n) / 2^(2*n-1).
%H G. C. Greubel, <a href="/A012250/b012250.txt">Table of n, a(n) for n = 1..225</a>
%H M. Hering and B. Howard, <a href="http://arxiv.org/abs/1211.3941">The ring of evenly weighted points on the line</a>, arXiv:1211.3941 [math.AG], 2012-2014, see p. 8.
%H B. Howard, J. Millson, A. Snowden, and R. Vakil, <a href="http://math.stanford.edu/~vakil/files/HMSV2aug1206.pdf">The moduli space of n points on the line is cut out by simple quadrics when n is not six</a>, p. 12.
%H Richard Stanley, <a href="http://mathoverflow.net/questions/123179/">Access to a preprint by D. N. Verma.</a>
%H R. P. Stanley and F. Zanello, <a href="http://math.mit.edu/~rstan/papers/distinctparts.pdf">Unimodality of partitions with distinct parts inside Ferrers shapes</a>, see Theorem 4.6.
%H R. P. Stanley and F. Zanello, <a href="http://arxiv.org/abs/1305.6083">Unimodality of partitions with distinct parts inside Ferrers shapes</a>, arXiv:1305.6083 [math.CO], 2013, see Theorem 4.6 and Remark 4.7.
%H R. P. Stanley and F. Zanello, <a href="https://doi.org/10.1016/j.ejc.2015.03.007">Unimodality of partitions with distinct parts inside Ferrers shapes</a>, European Journal of Combinatorics, Volume 49, October 2015, Pages 194-202.
%H 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, 1997, Unpublished.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 04 2021]
%F a(n) = (1/2)*Sum_{j=0..n} (-1)^(j+1)*binomial(2*n+2,j)*(n-j+1)^(2*n-1). - _Richard Stanley_, Mar 31 2013
%F a(n) ~ 3^(3/2) * 2^(2*n) * n^(2*n-2) / exp(2*n). - _Vaclav Kotesovec_, Oct 07 2021
%p A012250 := n -> 1/2*add((-1)^(j+1)*binomial(2*n+2,j)*(n-j+1)^(2*n-1)*(2*j-2*n-1),j=0..n); seq(A012250(i),i=1..9); # _Peter Luschny_, Mar 03 2013
%t Table[Sum[(-1)^(j + 1)*Binomial[2*n + 2, j]*(n - j + 1)^(2*n - 1)/2, {j, 0, n}], {n, 15}] (* _Wesley Ivan Hurt_, Nov 11 2014 *)
%o (Magma)
%o A012250:= func< n | (&+[(-1)^(j+1)*Binomial(2*n+2,j)*(n-j+1)^(2*n-1) : j in [0..n]])/2 >;
%o [A012250(n): n in [1..20]]; // _G. C. Greubel_, Feb 27 2024
%o (SageMath)
%o def A012250(n): return sum( (-1)^(j+1)*binomial(2*n+2,j)*(n-j+1)^(2*n-1) for j in range(n+1))/2
%o [A012250(n) for n in range(1,21)] # _G. C. Greubel_, Feb 27 2024
%Y Cf. A012249.
%K nonn
%O 1,2
%A _D n Verma_
%E Edited and extended using Richard Stanley's formula. - _N. J. A. Sloane_, Jun 10 2013
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