A185327 - OEIS

%I #44 Dec 01 2024 10:03:26

%S 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,

%T 27,32,36,42,48,56,63,73,83,96,108,125,141,162,183,209,236,270,304,

%U 346,390,443,498,565,635,719,807,911,1022,1153,1291,1453,1628,1829,2045

%N Number of partitions of n into parts >= 7.

%C a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 7 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to 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. Hence this sequence is essentially the same as A026800.

%H G. C. Greubel, <a href="/A185327/b185327.txt">Table of n, a(n) for n = 0..1000</a>

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_k-reg_girth_ge_g_index">Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g</a>

%F G.f.: Product_{m>=7} 1/(1-x^m).

%F a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + 2*p(n-7) - p(n-9) - p(n-10) - p(n-11) - p(n-12) + 2*p(n-14) + p(n-16) - p(n-19) - p(n-20) + p(n-21) where p(n)=A000041(n). - _Shanzhen Gao_, Oct 28 2010 [moved/copied from A026800 by _Jason Kimberley_, Feb 03 2011]

%F This sequence is the Euler transformation of A185117.

%F a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - _Vaclav Kotesovec_, Jun 02 2018

%F G.f.: Sum_{k>=0} x^(7*k) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Nov 28 2020

%F G.f.: 1 + Sum_{n >= 1} x^(n+6)/Product_{k = 0..n-1} (1 - x^(k+7)). - _Peter Bala_, Dec 01 2024

%e The  a(0)=1 empty partition vacuously has each part >= 7.

%e The  a(7)=1 partition is 7.

%e The  a(8)=1 partition is 8.

%e ............................

%e The a(13)=1 partition is 13.

%e The a(14)=2 partitions are 7+7 and 14.

%p seq(coeff(series(1/mul(1-x^(m+7), m = 0..80), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Nov 03 2019

%t 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]]]]; Table[ f[n, 7], {n, 0, 65}] (* _Robert G. Wilson v_, Jan 31 2011 *) (* moved from A026800 by _Jason Kimberley_, Feb 03 2011 *)

%t Join[{1},Table[Count[IntegerPartitions[n],_?(Min[#]>=7&)],{n,0,70}]] (* _Harvey P. Dale_, Oct 16 2011 *)

%t CoefficientList[Series[1/QPochhammer[x^7, x], {x, 0, 75}], x] (* _G. C. Greubel_, Nov 03 2019 *)

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

%o A185327 := func< n | p(n)-p(n-1)-p(n-2)+p(n-5)+2*p(n-7)-p(n-9)-p(n-10)- p(n-11)-p(n-12)+2*p(n-14)+p(n-16)-p(n-19)-p(n-20)+p(n-21) >;

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+7): m in [0..80]]) )); // _G. C. Greubel_, Nov 03 2019

%o (PARI) my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+7))) \\ _G. C. Greubel_, Nov 03 2019

%o (Sage)

%o def A185327_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( 1/product((1-x^(m+7)) for m in (0..80)) ).list()

%o A185327_list(70) # _G. C. Greubel_, Nov 03 2019

%Y 2-regular simple graphs with girth at least 7: A185117 (connected), A185227 (disconnected), this sequence (not necessarily connected).

%Y 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), this sequence (g=7), A185328 (g=8), A185329 (g=9).

%Y Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

%K nonn,easy

%O 0,15

%A _Jason Kimberley_, Feb 03 2011