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A081623 Number of ways in which the points on an n X n square lattice can be equally occupied with spin "up" and spin "down" particles. If n is odd, we arbitrarily take the lattice to contain one more spin "up" particle than the number of spin "down" particles. 1
1, 1, 6, 126, 12870, 5200300, 9075135300, 63205303218876, 1832624140942590534, 212392290424395860814420, 100891344545564193334812497256, 191645966716130525165099506263706416, 1480212998448786189993816895482588794876100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..57

Brian Hayes, The World in a Spin, American Scientist 88:5 (September-October 2000), pp. 384-388. [alternate link]

James Grime, Maths Problem: Complete Noughts and Crosses (Burnside's Lemma)

FORMULA

a(n) = C(n^2, (n^2+1)/2) if n is odd and C(n^2, n^2/2) if n is even.

a(n) = binomial(n^2,floor(n^2/2)). - Alois P. Heinz, Jul 21 2017

EXAMPLE

a(2) = C(4,2) = 6.

a(3) = C(9,5) = 126.

MAPLE

a:= n-> (s-> binomial(s, floor(s/2)))(n^2):

seq(a(n), n=0..15);  # Alois P. Heinz, Jul 21 2017

PROG

(Mathcad or Microsoft Excel): f(n)=combin(n^2, trunc((n^2+1)/2))

(PARI) a(n)=binomial(n^2, n^2\2) \\ Charles R Greathouse IV, May 09 2013

CROSSREFS

A082963 is the equivalent sequence up to reflection and rotation.

Sequence in context: A237428 A255900 A133792 * A223210 A177756 A089314

Adjacent sequences:  A081620 A081621 A081622 * A081624 A081625 A081626

KEYWORD

easy,nonn

AUTHOR

A. Timothy Royappa, Apr 22 2003

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Jul 21 2017

STATUS

approved

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Last modified September 20 08:44 EDT 2018. Contains 315232 sequences. (Running on oeis4.)