%I
%S 0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,4,4,5,6,7,8,10,11,
%T 13,15,18,20,24,27,32,36,42,48,56,63,73,83,96,108,125,141,162,183,209,
%U 236,270,304,346,390,443,498,565,635,719,807,911,1022,1153,1291,1453,1628,1829,2045
%N Number of partitions of n in which the least part is 7.
%C Contribution by _Jason Kimberley_, Feb 03 2011: (Start)
%C a(n) is also the number of not necessarily connected 2regular graphs on nvertices with girth exactly 7 (all such graphs are simple). The integer i corresponds to the icycle; the addition of integers corresponds to the disconnected union of cycles.
%C By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n  7. (End)
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_kreg_girth_eq_g_index">Index of sequences counting not necessarily connected kregular simple graphs with girth exactly g</a>
%F G.f.: x^7 * Product 1/(1x^m); m=7..inf.
%F a(n) = p(n7)p(n8)p(n9)+p(n12)+2*p(n14)p(n16)p(n17)p(n18)p(n19)+2*p(n21)+p(n23)p(n26)p(n27)+p(n28) where p(n)=A000041(n) including the implicit p(n)=0 for negative n. [From _Shanzhen Gao_, Oct 28 2010]  offset corrected / made explicit by _Jason Kimberley_, Feb 03 2011.
%e a(0)=0 because there does not exist a least part of the empty partition.
%e The a(7)=1 partition is 7.
%e The a(14)=1 partition is 7+7.
%e The a(15)=1 partition is 7+8.
%e .............................
%e The a(20)=1 partition is 7+13.
%e The a(21)=2 partitions are 7+7+7 and 7+14.
%o (MAGMA) p := func< n  n lt 0 select 0 else NumberOfPartitions(n) >;
%o A026800 := func< n  p(n7)p(n8)p(n9)+p(n12)+2*p(n14)p(n16) p(n17)p(n18)p(n19)+2*p(n21)+p(n23)p(n26)p(n27)+p(n28) >; // _Jason Kimberley_, Feb 03 2011
%Y Cf. A185327 (Mathematica code)
%Y 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), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
%Y Not necessarily connected 2regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2  multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), this sequence (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).  _Jason Kimberley_, Feb 03 2011
%K nonn,easy
%O 0,22
%A _Clark Kimberling_
%E More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001
