A019318 formula
David Wilson
davidwwilson at attbi.com
Thu May 29 19:02:29 CEST 2003
Per A019318:
I used Burnside's Lemma, as suggested, to compute a formula for
A019318. It was fairly easy once the application of the Lemma to
the problem in question was explained. The formula is still
somewhat ugly, and I used some cheats, but here goes.
Let
C(n, k) = choose
[n] = floor
n mod k = remainder
Define
A(n) = C(n^2, n)
B(n) = C((n^2-(n mod 2))/2, n/2);
C(n) = C((n^2-(n mod 2))/4, n/4);
D(n) = SUM(p = 0 to [n/2], C((n^2-n)/2, p)*C(n, n-2p))
Then
a(n) =
(A(n) + 3B(n) + 2C(n) + 2D(n))/8 if n == 0 (mod 4)
(A(n) + B(n) + 2C(n) + 4D(n))/8 if n == 1 (mod 4)
(A(n) + 3B(n) + 2D(n))/8 if n == 2 (mod 4)
(A(n) + B(n) + 4D(n))/8 if n == 3 (mod 4)
I used this formula to confirm and extend A019318 to full length.
I have submitted the terms, but not the formula, due to its length.
Maybe someone could submit the formula in an acceptable form,
it it is deemed worthy.
More information about the SeqFan
mailing list