A369628 Number of solutions to k_1 + 2*k_2 + ... + n*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

0, 1, 2, 3, 6, 15, 36, 85, 213, 549, 1423, 3723, 9882, 26508, 71579, 194533, 532120, 1463561, 4044075, 11221727, 31260192, 87386579, 245058185, 689209348, 1943530845, 5494106583, 15566303698, 44196212866, 125727934145, 358317169828, 1022916667066, 2924843243594

Offset: 0

Author

Ilya Gutkovskiy, Jan 28 2024

Keywords

• nonn

Crossrefs

Cf. A007576, A063866, A316706.

Programs

Mathematica

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

Python

from itertools import count, islice
from collections import Counter
def A369628_gen(): # generator of terms
    ccount = Counter({0:1})
    yield 0
    for i in count(1):
        bcount = Counter(ccount)
        for a in ccount:
            bcount[a+i] += ccount[a]
            bcount[a-i] += ccount[a]
        ccount = bcount
        yield(ccount[1])
A369628_list = list(islice(A369628_gen(),20)) # Chai Wah Wu, Jan 29 2024

Formula

a(n) = [x^1] Product_{k=1..n} (x^k + 1 + 1/x^k).

a(n) = [x^(n*(n+1)/2+1)] Product_{k=1..n} (1 + x^k + x^(2*k)).