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

Author

D n Verma

Keywords

• nonn

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• D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished.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

Crossrefs

Cf. A012249.

Programs

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

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 *)

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

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

Extensions

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