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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.

%I A334305 #23 Apr 24 2020 07:45:26
%S A334305 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,
%T A334305 30,34,39,44,48,56,62,69,78,88,96,110,122,134,152,168,186,208,231,254,
%U A334305 284,314,346,384,425,466,518,570,626,692,762,834,922,1010
%N 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.
%C A334305 a(n)>0 if n>=2k>=4.
%H A334305 Vaclav Kotesovec, <a href="/A334305/b334305.txt">Table of n, a(n) for n = 0..10000</a>
%F A334305 G.f.: Sum_{k>=2} x^(2k) Product_{i=2..k} (1+x^i).
%F A334305 From _Vaclav Kotesovec_, Apr 24 2020: (Start)
%F A334305 For n>=3, a(n) + a(n+1) = A087897(n+4).
%F A334305 a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (16 * 3^(3/4) * n^(5/4)). (End)
%e A334305 a(4)=1 because we have [2,2]; a(6)=2 because we have [2,2,2] and [3,3].
%e A334305 G.f.= x^4+2x^6+2x^8+x^9+2x^10+2x^11+3x^12+2x^13+...
%t A334305 nmax = 100; CoefficientList[Series[Sum[x^(2 k) Product[(1 + x^i), {i, 2, k}], {k, 2, nmax/2}], {x, 0, nmax}], x]
%t A334305 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 *)
%Y A334305 Cf. A000009, A000041, A025147, A087897.
%K A334305 nonn,easy
%O A334305 0,7
%A A334305 Victor Mishnyakov, _Elena Lanina_, Apr 22 2020