A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

1, -1, -3, -6, -18, -27, -108, -81, -486, 0, -1458, 8748, -6561, 118098, 118098, 1003833, 1417176, 11691702, 9565938, 105225318, 114791256, 746143164, 1076168025, 7231849128, 2324522934, 58113073350, 45328197213, 334731302496, 146444944842, 3263630199336, -3012581722464

Offset: 0

Author

Ilya Gutkovskiy, Jun 08 2022

Keywords

• sign

Crossrefs

Cf. A003056, A032308, A242587, A261582, A292128, A300579, A337299, A344062, A352762.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[(1 - 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 30}]

Formula

a(n) = Sum_{k=0..A003056(n)} (-1)^k * q(n,k) * 3^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.