login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A026799 Number of partitions of n in which the least part is 6. 19
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 21, 24, 29, 32, 40, 44, 53, 60, 71, 80, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 552, 634, 716, 819, 922, 1056, 1187, 1353, 1523, 1730, 1944, 2209, 2478, 2806, 3151 (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 exactly 6 (all such graphs are simple). Each integer part i corresponds to an i-cycle; the addition of integers corresponds to the disconnected union of cycles.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

FORMULA

G.f.: x^6 * Product_{m>=6} 1/(1-x^m).

a(n) = p(n-6) -p(n-7) -p(n-8) +p(n-11) +p(n-12) +p(n-13) -p(n-14) -p(n-15) -p(n-16) +p(n-19) +p(n-20) -p(n-21) for n>0 where p(n) = A000041(n). - Shanzhen Gao, Oct 28 2010

a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jun 02 2018

EXAMPLE

a(0)=0 because there does not exist a least part of the empty partition.

The  a(6)=1 partition is 6.

The a(12)=1 partition is 6+6.

The a(13)=1 partition is 6+7.

.............................

The a(17)=1 partition is 6+11.

The a(18)=2 partitions are 6+6+6 and 6+12.

MAPLE

ZL := [ B, {B=Set(Set(Z, card>=6))}, unlabeled ]: 0, 0, 0, 0, 0, 0, seq(combstruct[count](ZL, size=n), n=0..63); # Zerinvary Lajos, Mar 13 2007

seq(coeff(series(x^6/mul(1-x^(m+6), m=0..70), x, n+1), x, n), n = 0..65); # G. C. Greubel, Nov 03 2019

MATHEMATICA

f[1, 1]=f[0, k_]=1; f[n_, k_]:= f[n, k] = If[n<0, 0, If[k>n, 0, If[k==n, 1, f[n, k+1] +f[n-k, k]]]]; Join[{0, 0, 0, 0, 0, 0}, Table[f[n, 6], {n, 0, 65}]] (* Robert G. Wilson v, Jan 31 2011 *)

CoefficientList[Series[x^6/QPochhammer[x^6, x], {x, 0, 70}], x] (* G. C. Greubel, Nov 03 2019 *)

PROG

(MAGMA) p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;

A026799 := func< n | p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)- p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) >; // Jason Kimberley, Feb 04 2011

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 60); [0, 0, 0, 0, 0, 0] cat Coefficients(R!( x^6/(&*[1-x^(m+6): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

(PARI) my(x='x+O('x^60)); concat([0, 0, 0, 0, 0, 0], Vec(x^6/prod(m=0, 70, 1-x^(m+6)))) \\ G. C. Greubel, Nov 03 2019

(Sage)

def A026799_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( x^6/product((1-x^(m+6)) for m in (0..70)) ).list()

A026799_list(65) # G. C. Greubel, Nov 03 2019

CROSSREFS

Essentially the same as A185326.

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), this sequence (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 04 2011

Sequence in context: A185228 A026824 A025149 * A185326 A238209 A210716

Adjacent sequences:  A026796 A026797 A026798 * A026800 A026801 A026802

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 23 02:33 EST 2020. Contains 332159 sequences. (Running on oeis4.)