cp's OEIS Frontend

A060005 Number of ways of partitioning the integers {1,2,..,4n} into two (unordered) sets such that the sums of parts are equal in both sets (parts in either set will add up to (4n)*(4n+1)/4). Number of solutions to {1 +- 2 +- 3 +- ... +- 4n=0}.

%I A060005 #42 Jul 01 2017 07:33:49
%S A060005 1,1,7,62,657,7636,93846,1199892,15796439,212681976,2915017360,
%T A060005 40536016030,570497115729,8110661588734,116307527411482,
%U A060005 1680341334827514,24435006625667338,357366669614512168,5253165510907071170
%N A060005 Number of ways of partitioning the integers {1,2,..,4n} into two (unordered) sets such that the sums of parts are equal in both sets (parts in either set will add up to (4n)*(4n+1)/4). Number of solutions to {1 +- 2 +- 3 +- ... +- 4n=0}.
%H A060005 Alois P. Heinz and Ray Chandler, <a href="/A060005/b060005.txt">Table of n, a(n) for n = 0..835</a> (terms < 10^1000, first 251 terms from Alois P. Heinz)
%H A060005 Steven R. Finch, <a href="/A000980/a000980.pdf">Signum equations and extremal coefficients</a>, February 7, 2009. [Cached copy, with permission of the author]
%H A060005 L. Sallows, M. Gardner, R. K. Guy and D. E. Knuth, <a href="http://www.jstor.org/stable/2690648">Serial isogons of 90 degrees</a>, Math. Mag. 64 (1991), 315-324.
%F A060005 a(0)=1 and a(n) is half the coefficient of q^0 in product((q^(-k)+q^k), k=1..4*n) for n >= 1.
%F A060005 For n>=1, a(n) = (1/Pi)*16^n*J(4n) where J(n) = integral(t=0, Pi/2, cos(t)cos(2t)...cos(nt)dt). - _Benoit Cloitre_, Sep 24 2006
%e A060005 a(1)=1 since there is only one way of partitioning {1,2,3,4} into two sets of equal sum, namely {1,4}, {2,3}.
%p A060005 b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
%p A060005       `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i), i-1) +b(n+i, i-1)))
%p A060005     end:
%p A060005 a:= n-> b(4*n, 4*n-1):
%p A060005 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 30 2011
%t A060005 b[n_, i_] := b[n, i] = Module[{m = i*(i+1)/2}, If[n > m, 0, If[n == m, 1, b[Abs[n-i], i-1] + b[n+i, i-1]]]]; a[n_] := b[4*n, 4*n-1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Sep 26 2013, translated from Alois P. Heinz's Maple program *)
%Y A060005 Cf. A060468, A007219, A107350, a(n)=A058377(4n), A227850.
%K A060005 nonn
%O A060005 0,3
%A A060005 _Roland Bacher_, Mar 15 2001
%E A060005 More terms from _Alois P. Heinz_, Oct 30 2011