

A185325


Number of partitions of n into parts >= 5.


22



1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 13, 15, 18, 21, 26, 30, 36, 42, 50, 58, 70, 80, 95, 110, 129, 150, 176, 202, 236, 272, 317, 364, 423, 484, 560, 643, 740, 847, 975, 1112, 1277, 1456, 1666, 1897, 2168, 2464, 2809, 3189, 3627, 4112, 4673
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,11


COMMENTS

a(n) is also the number of not necessarily connected 2regular graphs on nvertices with girth at least 5 (all such graphs are simple). The integer i corresponds to the icycle; addition of integers corresponds to disconnected union of cycles.
By removing a single part of size 5, an A026798 partition of n becomes an A185325 partition of n  5. Hence this sequence is essentially the same as A026798.
a(n) = number of partitions of n+4 such that 4*(number of parts) is a part.  Clark Kimberling, Feb 27 2014


LINKS

Table of n, a(n) for n=0..60.
Jason Kimberley, Index of sequences counting not necessarily connected kregular simple graphs with girth at least g


FORMULA

G.f.: Product 1/(1x^m); m=5..inf.
Given by p(n)p(n1)p(n2)+2p(n5)p(n8)p(n9)+p(n10), where p(n) = A000041(n).  Shanzhen Gao, Oct 28 2010 [sign of 10 corrected from + to , and moved from A026798 to this sequence by Jason Kimberley].
This sequence is the Euler transformation of A185115.
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (6*sqrt(3)*n^3).  Vaclav Kotesovec, Jun 02 2018
G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1  x^k))).  Ilya Gutkovskiy, Aug 21 2018


MATHEMATICA

Drop[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*Length[p]]], {n, 40}], 3] (* Clark Kimberling, Feb 27 2014 *)


PROG

(MAGMA)
p := func< n  n lt 0 select 0 else NumberOfPartitions(n) >;
A185325 := func<n  p(n)p(n1)p(n2)+2*p(n5)p(n8)p(n9)+p(n10)>;
[A185325(n):n in[0..60]];


CROSSREFS

2regular simple graphs with girth at least 5: A185115 (connected), A185225 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 2regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1  multigraphs with loops allowed), A002865 (g=2  multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), this sequence (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected kregular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: this sequence (k=2), A185335 (k=3).
A008484, A008483.
Sequence in context: A036821 A237980 A026798 * A125890 A067661 A210024
Adjacent sequences: A185322 A185323 A185324 * A185326 A185327 A185328


KEYWORD

nonn,easy


AUTHOR

Jason Kimberley, Nov 11 2011


STATUS

approved



