A334305 a(n) is the number of partitions of n of the form [k,k,b(1),b(2),...], where k>=b(1)>b(2)>...>=2.

0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 2, 3, 2, 4, 4, 4, 6, 6, 7, 8, 10, 10, 13, 14, 16, 18, 22, 22, 28, 30, 34, 39, 44, 48, 56, 62, 69, 78, 88, 96, 110, 122, 134, 152, 168, 186, 208, 231, 254, 284, 314, 346, 384, 425, 466, 518, 570, 626, 692, 762, 834, 922, 1010

Offset: 0

Author

Victor Mishnyakov, Elena Lanina, Apr 22 2020

Keywords

• nonn
• easy

Comments

a(n)>0 if n>=2k>=4.

Examples

			a(4)=1 because we have [2,2]; a(6)=2 because we have [2,2,2] and [3,3].
G.f.= x^4+2x^6+2x^8+x^9+2x^10+2x^11+3x^12+2x^13+...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000009, A000041, A025147, A087897.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^(2 k) Product[(1 + x^i), {i, 2, k}], {k, 2, nmax/2}], {x, 0, nmax}], x]
Flatten[{{0, 0, 0}, Table[PartitionsQ[n + 3] - 2*(-1)^n + 2*Sum[(-1)^k * PartitionsQ[n - k + 3], {k, 1, n - 2}], {n, 3, 70}]}] (* Vaclav Kotesovec, Apr 24 2020 *)

Formula

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

From Vaclav Kotesovec, Apr 24 2020: (Start)

For n>=3, a(n) + a(n+1) = A087897(n+4).

a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (16 * 3^(3/4) * n^(5/4)). (End)