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A185329 Number of partitions of n with parts >= 9. 18
1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 24, 26, 30, 34, 39, 43, 50, 55, 63, 71, 80, 89, 102, 113, 128, 143, 161, 179, 203, 225, 253, 282, 316, 351, 395, 437, 489, 544, 607, 673, 752, 832, 927, 1028, 1143 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,19

COMMENTS

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 9 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.

By removing a single part of size 9, an A026802 partition of n becomes an A185329 partition of n - 9. Hence this sequence is essentially the same as A026802.

In general, if g>=1 and g.f. = Product_{m>=g} 1/(1-x^m), then a(n,g) ~ Pi^(g-1) * (g-1)! * exp(Pi*sqrt(2*n/3)) / (2^((g+3)/2) * 3^(g/2) * n^((g+1)/2)) ~ p(n) * Pi^(g-1) * (g-1)! / (6*n)^((g-1)/2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, Jun 02 2018

LINKS

Table of n, a(n) for n=0..70.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

FORMULA

G.f.: Product 1/(1-x^m); m=9..inf.

a(n) = p(n)-p(n-1)-p(n-2)+p(n-5)+p(n-7)+p(n-9)-p(n-11)-2p(n-12)-p(n-13)-p(n-15)+p(n-16)+p(n-17)+2p(n-18)+p(n-19)+p(n-20)-p(n-21)-p(n-23)-2p(n-24)-p(n-25)+p(n-27)+p(n-29)+p(n-31)-p(n-34)-p(n-35)+p(n-36) where p(n)=A000041(n). [From Shanzhen Gao].

This sequence is the Euler transformation of A185119.

a(n) ~ exp(Pi*sqrt(2*n/3)) * 70*Pi^8 / (9*sqrt(3)*n^5). - Vaclav Kotesovec, Jun 02 2018

CROSSREFS

Not necessarily connected 2-regular 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), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), this sequence (g=9).

Not necessarily connected 2-regular 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).

Sequence in context: A026827 A025152 A026802 * A029031 A188666 A029156

Adjacent sequences:  A185326 A185327 A185328 * A185330 A185331 A185332

KEYWORD

nonn,easy

AUTHOR

Jason Kimberley, Feb 01 2012

STATUS

approved

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Last modified July 23 20:32 EDT 2019. Contains 325264 sequences. (Running on oeis4.)