A318029 Expansion of Sum_{k>=2} x^(k*(k+3)/2) / Product_{j=1..k} (1 - x^j).

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 9, 11, 14, 16, 20, 24, 28, 34, 40, 47, 55, 65, 75, 88, 102, 118, 136, 158, 180, 208, 238, 272, 311, 355, 403, 459, 521, 590, 668, 756, 852, 962, 1084, 1218, 1370, 1538, 1724, 1932, 2163, 2417, 2701, 3015, 3361, 3745, 4170, 4636, 5154, 5724

Offset: 0

Author

Ilya Gutkovskiy, Aug 13 2018

Keywords

• nonn

Comments

Number of partitions of n into at least two distinct parts >= 2.

Examples

			a(9) = 4 because we have [7, 2], [6, 3], [5, 4] and [4, 3, 2].
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000009, A025147, A083751, A096765, A111133.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Sum[x^(k (k + 3)/2)/Product[(1 - x^j), {j, 1, k}], {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[x - 1/(1 - x) + 1/((1 + x) QPochhammer[x, x^2]), {x, 0, nmax}], x]
Join[{0, 0}, Table[-1 + Sum[(-1)^(n - k) PartitionsQ[k], {k, 0, n}], {n, 2, 60}]]

Formula

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

a(n) = A025147(n) - 1 for n > 1.