A104456 - OEIS

%I #26 Apr 03 2014 18:37:00

%S 1,4,35,361,4110,49910,632602,8273610,110826888,1512776590,

%T 20965992017,294245741167,4173319332859,59723919552183,

%U 861331863890066,12505857230438737,182650875111521033,2681644149792639400,39555354718945873299,585903163431438401072

%N Number of ways of partitioning the integers {1,2,..,4n-1} into two unordered sets such that the sums of parts are equal in both sets (parts in one of the sets hence sum up to n*(4n-1)). Number of solutions to {1 +- 2 +- 3+ ... +- 4n-1 = 0}.

%H Alois P. Heinz and Ray Chandler, <a href="/A104456/b104456.txt">Table of n, a(n) for n = 1..835</a> (terms < 10^1000, first 250 terms from Alois P. Heinz)

%F a(n) = A058377(4n-1). - _N. J. A. Sloane_, Jan 24 2006

%F a(n) is half the coefficient of q^(n*(4n - 1)) in the product('1 + x^j', 'j'=1..4*n-1), for n >= 1. - _N. J. A. Sloane_, Feb 24 2006

%F a(n) = (1/Pi)*2^(4n-1)*J(4n-1) where J(n) = integral(t=0, Pi/2, cos(t) * cos(2t) * ... * cos(nt)dt), n>=1. - _Benoit Cloitre_, Sep 24 2006

%F a(n) = A123117(n)/2. - _N. J. A. Sloane_, Jan 09 2009

%e a(2) = 4 since there are 4 ways of partitioning {1,2,3,4,5,6,7} into two sets of equal sum, namely {{1,2,5,6}, {3,4,7}}, {{1,3,4,6}, {2,5,7}}, {{2,3,4,5}, {1,6,7}}, {{1,2,4,7}, {3,5,6}}.

%e G.f. = x + 4*x^2 + 35*x^3 + 361*x^4 + 4110*x^5 + 49910*x^6 + ...

%p b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;

%p       `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i), i-1) +b(n+i, i-1)))

%p     end:

%p a:= n-> b(4*n-1, 4*n-2):

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Nov 01 2011

%t Table[CoefficientList[Product[1 + x^j, {j, 1, 4n - 1}], x][[n*(4n - 1) + 1]]/2, {n, 20}]

%Y Cf. A060005.

%K nonn

%O 1,2

%A Yiu Tung Poon (ytpoon(AT)iastate.edu) and Chun Chor Litwin Cheng (cccheng(AT)ied.edu.hk), Mar 08 2005