A369714 Number of solutions to 1^4*k_1 + 2^4*k_2 + ... + n^4*k_n = 1, where k_i are from {-1,0,1}, i=1..n.
0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 13, 25, 56, 110, 218, 494, 1216, 2702, 6477, 14752, 35758, 83730, 208107, 499459, 1250815, 3048590, 7787399, 19260830, 49686365, 124430675, 324018684, 820906005, 2155194085, 5514650519, 14578030389, 37630395887, 100201473164
Offset: 0
Keywords
Programs
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Maple
b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1, b(n, i-1)+ b(abs(n-i^4), i-1)+b(n+i^4, i-1))))(i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30) end: a:= n-> b(1, n): seq(a(n), n=0..33); # Alois P. Heinz, Jan 30 2024 -
Mathematica
b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[n, i-1] + b[Abs[n-i^4], i-1] + b[n+i^4, i-1]]]][i*(i+1)*(2*i+1)*(3*i^2 + 3*i-1)/30]; a[n_] := b[1, n]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 27 2025, after Alois P. Heinz *)
Formula
a(n) = [x^1] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).
Extensions
a(34)-a(37) from Alois P. Heinz, Jan 30 2024