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 A026800 Number of partitions of n in which the least part is 7. 18
 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, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,22 COMMENTS From Jason Kimberley, Feb 03 2011: (Start) a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 7 (all such graphs are simple). The integer i corresponds to the i-cycle; the addition of integers corresponds to the disconnected union of cycles. By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. (End) LINKS Robert Israel, Table of n, a(n) for n = 0..10000 FORMULA G.f.: x^7 * Product 1/(1-x^m); m=7..inf. a(n) = p(n-7)-p(n-8)-p(n-9)+p(n-12)+2*p(n-14)-p(n-16)-p(n-17)-p(n-18)-p(n-19)+2*p(n-21)+p(n-23)-p(n-26)-p(n-27)+p(n-28) where p(n)=A000041(n) including the implicit p(n)=0 for negative n. [Shanzhen Gao, Oct 28 2010; offset corrected / made explicit by Jason Kimberley, Feb 03 2011] a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - Vaclav Kotesovec, Jun 02 2018 EXAMPLE a(0)=0 because there does not exist a least part of the empty partition. The  a(7)=1 partition is 7. The a(14)=1 partition is 7+7. The a(15)=1 partition is 7+8. ............................. The a(20)=1 partition is 7+13. The a(21)=2 partitions are 7+7+7 and 7+14. MAPLE N:= 100: # for a(0)..a(N) S:= series(x^7/mul(1-x^i, i=7..N-7), x, N+1): seq(coeff(S, x, i), i=0..N); # Robert Israel, Jul 04 2019 PROG (MAGMA) p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >; A026800 := func< n | p(n-7)-p(n-8)-p(n-9)+p(n-12)+2*p(n-14)-p(n-16)- p(n-17)-p(n-18)-p(n-19)+2*p(n-21)+p(n-23)-p(n-26)-p(n-27)+p(n-28) >; // Jason Kimberley, Feb 03 2011 CROSSREFS Cf. A185327 (Mathematica code) 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), A185329 (g=9). Not necessarily connected 2-regular 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 Sequence in context: A185229 A026825 A025150 * A185327 A210717 A171962 Adjacent sequences:  A026797 A026798 A026799 * A026801 A026802 A026803 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001 STATUS approved

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Last modified September 23 02:37 EDT 2019. Contains 327327 sequences. (Running on oeis4.)