A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).

1, 1, 1, 1, 3, 5, 11, 27, 61, 133, 311, 761, 1839, 4575, 11573, 29641, 76487, 199617, 524067, 1384697, 3681069, 9841217, 26437741, 71369101, 193496241, 526685793, 1438816755, 3944034221, 10845006963, 29908325821, 82707648985, 229306378067, 637283978821

Offset: 0

Author

Ilya Gutkovskiy, Jan 20 2024

Keywords

• nonn

Comments

All terms are odd.

a(n) is the number of solutions to 0 = Sum_{i=1..n} c_i * i*(i+1)/2 with c_i in {-1,0,1}.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000217, A007576, A158380, A350249.

Programs

Maple

b:= proc(n, i) option remember; `if`(n>i*(i+1)*(i+2)/6, 0, `if`(i=0, 1,
      b(n, i-1)+b(n+i*(i+1)/2, i-1)+b(abs(n-i*(i+1)/2), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 21 2024

Mathematica

Table[Coefficient[Product[x^(k (k + 1)/2) + 1 + 1/x^(k (k + 1)/2), {k, 1, n}], x, 0], {n, 0, 31}]

Formula

a(n) ~ sqrt(5) * 3^(n + 1/2) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Jan 22 2024