A341251 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^9.

1, 0, 9, 9, 45, 81, 201, 414, 828, 1650, 3060, 5697, 10131, 17829, 30564, 51543, 85482, 139455, 224527, 356436, 559341, 867405, 1331208, 2022525, 3044331, 4542174, 6720705, 9866794, 14377941, 20804994, 29903823, 42709860, 60631011, 85575855, 120118500, 167716548

Offset: 9

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..10000

Crossrefs

Cf. A000700, A001487, A022604, A327387, A338463, A341228, A341241, A341243, A341244, A341245, A341246, A341247.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..44);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

Formula

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^9.