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

1,19

Comments

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

Links

Table of n, a(n) for n=1..58.

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

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

Crossrefs

Cf. A058377, A083527, A113263.

Sequence in context: A213153 A132038 A087495 * A021533 A073242 A217597

Adjacent sequences: A111250 A111251 A111252 * A111254 A111255 A111256

Keyword

nonn

Author

Robert G. Wilson v, Oct 31 2005

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

Status

approved