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.

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, 1396, 2994, 0, 0, 14529, 22553, 0, 0, 138414, 236927, 0, 0, 1227670, 2388718, 0, 0, 13733162, 23214820, 0, 0, 140197641, 263244668, 0, 0, 1596794975, 2830613464, 0, 0

Offset: 1

Author

Robert G. Wilson v, Oct 31 2005

Keywords

• nonn

Comments

a(n)=0 when n == 1 or 2 (mod 4).

Crossrefs

Cf. A058377, A083527, A113263.

Programs

Maple

b:= proc(n, i) option remember; local m;
      m:= (-1+(10+(15+6*i)*i)*i^2)*i/30;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^4), i-1) +b(n+i^4, i-1)))
    end:
a:= n-> `if`(irem(n-1, 4)<2, 0, b(n^4, n-1)):
seq(a(n), n=1..38);  # Alois P. Heinz, Oct 30 2011

Mathematica

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 *)
p = 1; t = {}; Do[p = Expand[p(x^(n^4) + x^(-n^4))]; AppendTo[t, Select[p, NumberQ[ # ] &]/2], {n, 30}]; t

Formula

a(n) is half the coefficient of x^0 in product_{k=1..n} x^(k^4)+x^(k^-4).

a(n) = [x^(n^4)] Product_{k=1..n-1} (x^(k^4) + 1/x^(k^4)). - Ilya Gutkovskiy, Feb 01 2024

Extensions

a(51)-a(54) from T. D. Noe, Nov 01 2005

Corrected a(51)-a(52) and extended up to a(58) by Alois P. Heinz, Oct 31 2011