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.
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
FORMULA
G.f.: x^7 * Product_{m>=7} 1/(1-x^m).
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
G.f.: Sum_{k>=1} x^(7*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020
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
MATHEMATICA
CoefficientList[Series[x^7/QPochhammer[x^7, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 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
(Magma) R<x>:=PowerSeriesRing(Integers(), 75); [0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!( x^7/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019
(PARI) my(x='x+O('x^75)); concat([0, 0, 0, 0, 0, 0, 0], Vec(x^7/prod(m=0, 80, 1-x^(m+7)))) \\ G. C. Greubel, Nov 03 2019
(Sage)
def A026800_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^7/product((1-x^(m+7)) for m in (0..80)) ).list()
A026800_list(75) # G. C. Greubel, Nov 03 2019
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001
STATUS
approved