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A111253 a(n) is the number of ways the set {1^4, 2^4, ..., n^4} can be partitioned into two sets of equal sums.

%I A111253 #19 Feb 01 2024 08:10:46
%S A111253 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,9,0,0,16,50,0,0,212,255,0,0,
%T A111253 1396,2994,0,0,14529,22553,0,0,138414,236927,0,0,1227670,2388718,0,0,
%U A111253 13733162,23214820,0,0,140197641,263244668,0,0,1596794975,2830613464,0,0
%N A111253 a(n) is the number of ways the set {1^4, 2^4, ..., n^4} can be partitioned into two sets of equal sums.
%C A111253 a(n)=0 when n == 1 or 2 (mod 4).
%F A111253 a(n) is half the coefficient of x^0 in product_{k=1..n} x^(k^4)+x^(k^-4).
%F A111253 a(n) = [x^(n^4)] Product_{k=1..n-1} (x^(k^4) + 1/x^(k^4)). - _Ilya Gutkovskiy_, Feb 01 2024
%p A111253 b:= proc(n, i) option remember; local m;
%p A111253       m:= (-1+(10+(15+6*i)*i)*i^2)*i/30;
%p A111253       `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^4), i-1) +b(n+i^4, i-1)))
%p A111253     end:
%p A111253 a:= n-> `if`(irem(n-1, 4)<2, 0, b(n^4, n-1)):
%p A111253 seq(a(n), n=1..38);  # _Alois P. Heinz_, Oct 30 2011
%t A111253 d = {1, 1}; nMax=50; zeroLst = {0}; Do[p = n^4; d = PadLeft[d, Length[d] + p] + PadRight[d, Length[d] + p]; If[1 == Mod[Length[d], 2], AppendTo[zeroLst, d[[(Length[d] + 1)/2]]], AppendTo[zeroLst, 0]], {n, 2, nMax}]; zeroLst/2 (* _T. D. Noe_, Oct 31 2005 *)
%t A111253 p = 1; t = {}; Do[p = Expand[p(x^(n^4) + x^(-n^4))]; AppendTo[t, Select[p, NumberQ[ # ] &]/2], {n, 30}]; t
%Y A111253 Cf. A058377, A083527, A113263.
%K A111253 nonn
%O A111253 1,19
%A A111253 _Robert G. Wilson v_, Oct 31 2005
%E A111253 a(51)-a(54) from _T. D. Noe_, Nov 01 2005
%E A111253 Corrected a(51)-a(52) and extended up to a(58) by _Alois P. Heinz_, Oct 31 2011