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).
0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 209, 267, 343, 435, 551, 693, 870, 1084, 1351, 1672, 2066, 2542, 3121, 3815, 4658, 5664, 6875, 8319, 10049, 12102, 14553, 17452, 20894, 24959, 29766, 35420, 42089, 49911, 59100, 69856, 82452, 97152, 114324, 134315
Offset: 1
Keywords
Examples
a(7) = 2 counts these partitions: 7, 6+1.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Mathematica
Table[Count[IntegerPartitions[n],?(#[[1]]-Length[#]>3&)],{n,60}] (* _Harvey P. Dale, Jul 29 2024 *)
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PARI
a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+7)/2));
Formula
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+7)/2).
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().
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
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