cp's OEIS Frontend

A363231 Number of partitions of n with rank 4 or higher (the rank of a partition is the largest part minus the number of parts).

%I A363231 #31 Jul 29 2024 12:20:19
%S A363231 0,0,0,0,1,1,2,3,5,7,11,15,21,29,40,53,72,94,124,161,209,267,343,435,
%T A363231 551,693,870,1084,1351,1672,2066,2542,3121,3815,4658,5664,6875,8319,
%U A363231 10049,12102,14553,17452,20894,24959,29766,35420,42089,49911,59100,69856,82452,97152,114324,134315
%N A363231 Number of partitions of n with rank 4 or higher (the rank of a partition is the largest part minus the number of parts).
%C A363231 In general, for r>=0, Sum_{k>=1} (-1)^(k-1) * p(n - k*(3*k + 2*r - 1)/2) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + (12*r-5)*Pi/144) /  sqrt(n/6)), where p() is the partition function. - _Vaclav Kotesovec_, May 26 2023
%H A363231 Vaclav Kotesovec, <a href="/A363231/b363231.txt">Table of n, a(n) for n = 1..10000</a>
%F A363231 G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+7)/2).
%F A363231 a(n) = p(n-5) - p(n-13) + p(n-24) - ... + (-1)^(k-1) * p(n-k*(3*k+7)/2) + ..., where p() is A000041().
%F A363231 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 43*Pi/144) / sqrt(n/6)). - _Vaclav Kotesovec_, May 26 2023
%e A363231 a(7) = 2 counts these partitions: 7, 6+1.
%t A363231 Table[Count[IntegerPartitions[n],_?(#[[1]]-Length[#]>3&)],{n,60}] (* _Harvey P. Dale_, Jul 29 2024 *)
%o A363231 (PARI) a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+7)/2));
%Y A363231 With rank r or higher: A064174 (r=0), A064173 (r=1), A123975 (r=2), A363230 (r=3), this sequence (r=4).
%Y A363231 Cf. A000041, A140090, A363213.
%K A363231 nonn
%O A363231 1,7
%A A363231 _Seiichi Manyama_, May 22 2023