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%I #31 Jul 01 2017 07:33:44
%S 1,2,14,124,1314,15272,187692,2399784,31592878,425363952,5830034720,
%T 81072032060,1140994231458,16221323177468,232615054822964,
%U 3360682669655028,48870013251334676,714733339229024336
%N Number of fair distributions (equal sum) of the integers {1,..,4n} between A and B = number of solutions to the equation {+-1 +-2 +- 3 ... +-4*n = 0}.
%H Ray Chandler, <a href="/A060468/b060468.txt">Table of n, a(n) for n = 0..834</a> (terms < 10^1000)
%H Steven R. Finch, <a href="/A000980/a000980.pdf">Signum equations and extremal coefficients</a>, February 7, 2009. [Cached copy, with permission of the author]
%F a(n) = coefficient of q^0 in Product_{k=1..4*n} (q^(-k) + q^k).
%F a(n) = A025591(4n) = A063865(4n) = A063867(4n) = 2*A060005(n). Seems to be close to sqrt(3/32Pi)*16^n/sqrt(n^3 + n^2*0.6 + n*0.1385...) and sqrt(n*Pi/2)*A063074(n). - _Henry Bottomley_, Jul 30 2005
%e a(1)=2: give either the set {1,4} to A and {2,3} to B or give {2,3} to A and {1,4} to B.
%t a[n_] := Coefficient[Product[q^(-k) + q^k, {k, 1, 4*n}], q, 0]; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Sep 26 2013 *)
%Y Cf. A025591, A060005, A063865, A063867.
%K nice,nonn
%O 0,2
%A _Roland Bacher_, Mar 15 2001
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