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A186445 The number of partitions of n in which the first part is at least four times larger than the second part.

%I A186445 #12 Jul 05 2025 05:52:38
%S A186445 1,1,1,1,1,2,3,4,5,6,8,10,13,16,20,25,31,38,47,57,70,85,103,124,150,
%T A186445 180,216,258,308,366,436,516,611,721,850,1000,1176,1378,1614,1886,
%U A186445 2203,2567,2990,3474,4034,4677,5417,6264,7239,8351,9628
%N A186445 The number of partitions of n in which the first part is at least four times larger than the second part.
%F A186445 a(n) = p(n)-p(n-2)-p(n-3)-p(n-4)+p(n-5)+p(n-6)+p(n-7)-p(n-9), where p(n) = A000041(n).
%F A186445 a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3*sqrt(2)*n^(5/2)) * (1 - (5*sqrt(6)/Pi + 109*Pi/(24*sqrt(6)))/sqrt(n)). - _Vaclav Kotesovec_, Jul 05 2025
%e A186445 a(8) = #{8, 7+1, 6+1+1, 5+1+1+1, 4+1+1+1+1} = 5.
%e A186445 a(10) = #{10, 9+1, 8+2, 8+1+1, 7+1+1+1, 6+1+1+1+1, 5+1+1+1+1+1, 4+1+1+1+1+1+1} = 8.
%t A186445 Table[PartitionsP[n] - PartitionsP[n-2] - PartitionsP[n-3] - PartitionsP[n-4] + PartitionsP[n-5] + PartitionsP[n-6] + PartitionsP[n-7] - PartitionsP[n-9], {n,0,50}] (* _Vaclav Kotesovec_, Jul 05 2025 *)
%Y A186445 Cf. A000041.
%Y A186445 Partial sums of A185325.
%K A186445 nonn
%O A186445 0,6
%A A186445 _Mircea Merca_, Feb 21 2011