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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. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

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

LINKS

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

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)

EXAMPLE

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+...

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 *)

CROSSREFS

Cf. A000009, A000041, A025147, A087897.

Sequence in context: A045450 A029222 A162350 * A125922 A283306 A182893

Adjacent sequences:  A334302 A334303 A334304 * A334306 A334307 A334308

KEYWORD

nonn,easy

AUTHOR

Victor Mishnyakov, Elena Lanina, Apr 22 2020

STATUS

approved

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Last modified February 25 14:40 EST 2021. Contains 341609 sequences. (Running on oeis4.)