OFFSET
0,13
COMMENTS
a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 6 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.
LINKS
Jason Kimberley, Table of n, a(n) for n = 0..998
FORMULA
G.f.: Product_{m>=6} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-6) + p(n-7) - p(n-8) - p(n-9) - p(n-10) + p(n-13) + p(n-14) - p(n-15) where p(n) = A000041(n).
This sequence is the Euler transformation of A185116.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(6*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
MAPLE
seq(coeff(series(1/mul(1-x^(m+6), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019
MATHEMATICA
CoefficientList[Series[1/QPochhammer[x^6, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)
PROG
(Magma) A185326 := func<n|#RestrictedPartitions(n, {6..n})>;
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+6): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019
(PARI) my(x='x+O('x^70)); Vec(1/prod(m=0, 80, 1-x^(m+6))) \\ G. C. Greubel, Nov 03 2019
(Sage)
def A185326_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/product((1-x^(m+6)) for m in (0..80)) ).list()
A185326_list(70) # G. C. Greubel, Nov 03 2019
CROSSREFS
2-regular simple graphs with girth at least 6: A185116 (connected), A185226 (disconnected), this sequence (not necessarily connected).
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), this sequence (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
KEYWORD
nonn,easy
AUTHOR
Jason Kimberley, Jan 30 2012
STATUS
approved