A185326 - OEIS

%I #30 Dec 01 2024 10:03:50

%S 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,

%T 40,44,53,60,71,80,96,107,126,143,167,188,221,248,288,326,376,424,491,

%U 552,634,716,819,922,1056,1187,1353,1523,1730,1944,2209,2478,2806,3151

%N Number of partitions of n into parts >= 6.

%C 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.

%C By removing a single part of size 6, an A026799 partition of n becomes an A185326 partition of n - 6. Hence this sequence is essentially the same as A026799.

%H Jason Kimberley, <a href="/A185326/b185326.txt">Table of n, a(n) for n = 0..998</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>=6} 1/(1-x^m).

%F 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).

%F a(n) = A185226(n) + A185116(n).

%F This sequence is the Euler transformation of A185116.

%F a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - _Vaclav Kotesovec_, Jun 02 2018

%F G.f.: Sum_{k>=0} x^(6*k) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Nov 28 2020

%F G.f.: 1 + Sum_{n >= 1} x^(n+5)/Product_{k = 0..n-1} (1 - x^(k+6)). - _Peter Bala_, Dec 01 2024

%p 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

%t CoefficientList[Series[1/QPochhammer[x^6, x], {x, 0, 75}], x] (* _G. C. Greubel_, Nov 03 2019 *)

%o (Magma) A185326 := func<n|#RestrictedPartitions(n,{6..n})>;

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+6): 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+6))) \\ _G. C. Greubel_, Nov 03 2019

%o (Sage)

%o def A185326_list(prec):

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

%o     return P( 1/product((1-x^(m+6)) for m in (0..80)) ).list()

%o A185326_list(70) # _G. C. Greubel_, Nov 03 2019

%Y 2-regular simple graphs with girth at least 6: A185116 (connected), A185226 (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), this sequence (g=6), A185327 (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 0,13

%A _Jason Kimberley_, Jan 30 2012