A012250 a(n) = A012249(2*n) / 2^(2*n-1).

1, 3, 40, 1225, 67956, 5986134, 769550496, 136151219061, 31753157473180, 9445432588519642, 3491687484842443536, 1570713950508131878618, 845034544811095556274280, 535857105694970626486925100, 395590680969537758258609408640, 336386798400777928783348084420365

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..225

M. Hering and B. Howard, The ring of evenly weighted points on the line, arXiv:1211.3941 [math.AG], 2012-2014, see p. 8.

B. Howard, J. Millson, A. Snowden, and R. Vakil, The moduli space of n points on the line is cut out by simple quadrics when n is not six, p. 12.

Richard Stanley, Access to a preprint by D. N. Verma.

R. P. Stanley and F. Zanello, Unimodality of partitions with distinct parts inside Ferrers shapes, see Theorem 4.6.

R. P. Stanley and F. Zanello, Unimodality of partitions with distinct parts inside Ferrers shapes, arXiv:1305.6083 [math.CO], 2013, see Theorem 4.6 and Remark 4.7.

R. P. Stanley and F. Zanello, Unimodality of partitions with distinct parts inside Ferrers shapes, European Journal of Combinatorics, Volume 49, October 2015, Pages 194-202.

D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished.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 04 2021]

Formula

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

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

Maple

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

Mathematica

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 *)

Prog

(Magma)
A012250:= func< n | (&+[(-1)^(j+1)*Binomial(2*n+2, j)*(n-j+1)^(2*n-1) : j in [0..n]])/2 >;
[A012250(n): n in [1..20]]; // G. C. Greubel, Feb 27 2024
(SageMath)
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
[A012250(n) for n in range(1, 21)] # G. C. Greubel, Feb 27 2024

Crossrefs

Cf. A012249.

Sequence in context: A260754 A047799 A204515 * A094330 A110468 A327356

Adjacent sequences: A012247 A012248 A012249 * A012251 A012252 A012253

Keyword

nonn

Author

D n Verma

Extensions

Edited and extended using Richard Stanley's formula. - N. J. A. Sloane, Jun 10 2013

Status

approved