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).
0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 120, 154, 201, 256, 330, 415, 529, 662, 833, 1035, 1293, 1595, 1976, 2425, 2982, 3640, 4449, 5401, 6565, 7935, 9592, 11543, 13891, 16645, 19943, 23808, 28408, 33792, 40172, 47619, 56413, 66661, 78708, 92724, 109149, 128213, 150486, 176293
Offset: 1
Keywords
Examples
a(6) = 2 counts these partitions: 6, 5+1.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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PARI
a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+5)/2));
Formula
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+5)/2).
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().
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