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%I #50 May 16 2022 04:38:28
%S 1,2,8,58,526,5448,61108,723354,8908546,113093022,1470597342,
%T 19499227828,262754984020,3589093760726,49596793134484,
%U 692260288169282,9747120868919060,138298900243896166,1975688102624819336,28396056820503468894,410363630540693436398
%N 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 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. 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_, May 02 2002
%C 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. Example for n=2: {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 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
%H Alois P. Heinz, <a href="/A063074/b063074.txt">Table of n, a(n) for n = 0..100</a>
%H H. Prodinger, <a href="http://dx.doi.org/10.4153/CMB-1982-034-5">On the number of partitions of 1...n into two sets of equal cardinalities and equal sums</a>, Canad. Math. Bull. 25(1982), pp. 238-241.
%F 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_, May 12 2005
%e 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).
%p b:= proc(n, i, t) option remember;
%p `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
%p end:
%p a:= n-> b(n*(4*n+1), 4*n, 2*n):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 18 2012
%t Table[ Length@Select[ IntegerPartitions[ 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.
%t (* second program: *)
%t b[n_, i_, t_] := b[n, i, t] = If[i < t || n < t (t + 1)/2 || n > t (2i - t + 1)/2, 0, If[n == 0, 1, b[n, i - 1, t] + If[n < i, 0, b[n - i, i - 1, t - 1]]]]; a[n_] := b[n (4n + 1), 4n, 2n]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%Y Cf. A047993, A063075.
%Y Bisection of row n=2 of A204459. - _Alois P. Heinz_, Jan 18 2012
%K nonn
%O 0,2
%A _Wouter Meeussen_, Aug 03 2001
%E More terms from _Alois P. Heinz_, Jan 18 2012
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