%I A063074
%S A063074 1,2,8,58,526,5448,61108,723354,8908546,113093022,1470597342,
%T A063074 19499227828,262754984020,3589093760726,49596793134484,692260288169282,
%U A063074 9747120868919060,138298900243896166,1975688102624819336
%N A063074 Number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n 
               box; number of ways to cut a 2n X 2n chessboard into two equal-area 
               pieces along a non-descending line from lower left to upper right.
%C A063074 Also the number of subsets of {1,..,4n} containing exactly 2n elements 
               with total sum n*(4n+1) (see also A060468 for a related sequence). 
               This is of course the same as the number of partitions of n*(4n+1) 
               having 2n distinct parts of length at most 4n. (cont.)
%C A063074 (cont.). This number is the coefficient of t^0 q^0 in the product('(t*q^k+1/
               (t*q^k)','k'=1..4*n). - Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), 
               May 02 2002
%C A063074 A bijection with a dissection as above of the 2n X 2n checkerboard is 
               given by subtracting 1,2,3,..,2n of the smallest, second-smallest, 
               etc. element in the subset.
%C A063074 Example (n=2 as above): {1,2,7,8} (yields the checkerboard partition 
               {1-1,2-2,7-3,8-4}={0,0,4,4}), {1,3,6,8} (yields {1-1,3-2,6-3,8-4}={0,
               1,3,4}), {1,4,5,8} (yields {0,2,2,4}), {1,4,6,7} (yields {0,2,3,3}), 
               {3,4,5,6} (yields {2,2,2,2}), {2,4,5,7} (yields {1,2,2,3}), {2,3,
               6,7} (yields {1,1,3,3}), {2,3,5,8} (yields {1,1,2,4}).
%C A063074 Appears to be the number of random walks of length 4n, moves +/-1, starting 
               and ending at 0 and with signed area 0 under the path. It would be 
               nice to have a lower bound of the form a(n) > c*2^{4n}/n^d - David_Mumford(AT)brown.edu, 
               Jun 25 2003
%F A063074 a(n) = A029895(2n) = A067059(2n, 2n) = A107110(2n, 2n^2). a(n) seems 
               to be close to (sqrt(75)/pi)*16^n/(20n^2+6n+1). - Henry Bottomley 
               (se16(AT)btinternet.com), May 12 2005
%e A063074 For a 4 X 4 board (n=2) the 8 partitions are (4, 4, 0, 0), (4, 3, 1, 
               0), (4, 2, 1, 1), (4, 2, 2, 0), (3, 3, 2, 0), (3, 3, 1, 1), (3, 2, 
               2, 1), (2, 2, 2, 2).
%t A063074 Table[ Length@Select[ Partitions[ 2n^2 ], Length[ # ] <= 2n && First[ 
               # ] <= 2n& ], {n, 1, 5} ] or faster: Table[ T[ 2n^2, 2n, 2n ], {n, 
               0, 24} ] with T[ m, a, b ] as defined in A047993.
%Y A063074 Cf. A047993, A063075.
%Y A063074 Sequence in context: A153525 A153553 A007347 this_sequence A005804 A162067 
               A086907
%Y A063074 Adjacent sequences: A063071 A063072 A063073 this_sequence A063075 A063076 
               A063077
%K A063074 nonn
%O A063074 0,2
%A A063074 Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 03 2001