cp's OEIS Frontend

A363230 Number of partitions of n with rank 3 or higher (the rank of a partition is the largest part minus the number of parts).

%I A363230 #26 May 26 2023 09:54:24
%S A363230 0,0,0,1,1,2,3,5,7,11,14,21,28,39,51,70,90,120,154,201,256,330,415,
%T A363230 529,662,833,1035,1293,1595,1976,2425,2982,3640,4449,5401,6565,7935,
%U A363230 9592,11543,13891,16645,19943,23808,28408,33792,40172,47619,56413,66661,78708,92724,109149,128213,150486,176293
%N A363230 Number of partitions of n with rank 3 or higher (the rank of a partition is the largest part minus the number of parts).
%H A363230 Vaclav Kotesovec, <a href="/A363230/b363230.txt">Table of n, a(n) for n = 1..10000</a>
%F A363230 G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+5)/2).
%F A363230 a(n) = p(n-4) - p(n-11) + p(n-21) - ... + (-1)^(k-1) * p(n-k*(3*k+5)/2) + ..., where p() is A000041().
%F A363230 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 31*Pi/144) / sqrt(n/6)). - _Vaclav Kotesovec_, May 26 2023
%e A363230 a(6) = 2 counts these partitions: 6, 5+1.
%o A363230 (PARI) a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+5)/2));
%Y A363230 With rank r or higher: A064174 (r=0), A064173 (r=1), A123975 (r=2), this sequence (r=3), A363231 (r=4).
%Y A363230 Cf. A000041, A101200, A115067.
%K A363230 nonn
%O A363230 1,6
%A A363230 _Seiichi Manyama_, May 22 2023