A392139 Number of ways to write n^4 as an ordered sum of n fourth powers of integers.

1, 2, 4, 6, 8, 970, 396, 13454, 16, 6644754, 610580, 374922262, 354840, 365358090266, 10420009628, 129376411644510, 37970205802528, 22202426383718434, 13471239507620723748, 514525165340135141414, 272732900529553785789480, 10232748544503830232201258, 3371776098865394678360612908

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = [x^(n^4)] ( Sum_{j=-oo..oo} x^(j^4) )^n.

Example

There are a(3) = 6 solutions (x,y,z) of 3^4 = x^4 + y^4 + z^4: (-3,0,0), (0,-3,0), (0,0,-3), (0,0,3), (0,3,0) and (3,0,0).

Mathematica

Table[SeriesCoefficient[Sum[x^(j^4), {j, -n, n}]^n, {x, 0, n^4}], {n, 0, 16}]

Prog

(PARI) a(n) = polcoef((sum(j=-n, n, x^(j^4)) + O(x^(n^4+1)))^n, n^4) \\ Jason Yuen, Jan 01 2026

Crossrefs

Cf. A000583, A175372, A175375, A232173, A259793, A299169, A299195, A307644, A346566.

Sequence in context: A351777 A030149 A083146 * A306091 A074108 A231408

Adjacent sequences: A392136 A392137 A392138 * A392140 A392141 A392142

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 01 2026

Extensions

a(17)-a(22) from Jason Yuen, Jan 01 2026

Status

approved