|
%I #89 Nov 09 2023 12:20:15
%S 1,0,0,0,1,1,1,1,2,2,3,3,5,5,7,8,11,12,16,18,24,27,34,39,50,57,70,81,
%T 100,115,140,161,195,225,269,311,371,427,505,583,688,791,928,1067,
%U 1248,1434,1668,1914,2223,2546,2945,3370,3889,4443,5113,5834,6698
%N Number of partitions of n into parts >= 4.
%C a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 4 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - _Jason Kimberley_, Jan 2011 and Feb 2012
%C By removing a single part of size 4, an A026797 partition of n becomes an A008484 partition of n - 4. Hence this sequence is essentially the same as A026797. - _Jason Kimberley_, Feb 2012
%C Number of partitions of n+3 such that 3*(number of parts) is a part. - _Clark Kimberling_, Feb 27 2014
%C Let c(n) be the number of partitions of n such that both (number of parts) and 2*(number of parts) are parts; then c(n) = a(n-6) for n >= 6 and c(n) = 0 for n < 6. - _Clark Kimberling_, Mar 01 2014
%C a(n) is also the number of partitions of n for which three times the number of ones is twice the number of parts (conjectured). - _George Beck_, Aug 19 2017
%C Proof: Above definition is equivalent to 2 out of 3 parts being equal to 1. Arrange in triples 1, 1, >= 2, etc. Sum of each triple corresponds to sequence definition. - _Martin Fuller_, Aug 21 2023
%H Giovanni Resta, <a href="/A008484/b008484.txt">Table of n, a(n) for n = 0..1000</a>
%H Kevin Beanland and Hung Viet Chu, <a href="https://arxiv.org/abs/2311.01926">On Schreier-type Sets, Partitions, and Compositions</a>, arXiv:2311.01926 [math.CO], 2023.
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_girth_ge_4">Not necessarily connected k-regular graphs with girth at least 4</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.: 1 / Product_{m>=4} (1 - x^m).
%F Euler transformation of A185114. - _Jason Kimberley_, Jan 30 2011
%F Given by p(n) - p(n-1) - p(n-2) + p(n-4) + p(n-5) - p(n-6) where p(n) = A000041(n). Generally, 1/Product_{i>=K} (1 - x^i) is given by p({A}), where {A} is defined over the coefficients of Product_{i=1..K-1} (1 - x^i). In this case, K=4, so (1-x)(1-x^2)(1-x^3) = 1 - x - x^2 + x^4 + x^5 - x^6, defining {A} as above. G.f.: 1 + Sum_{i>=1} (x^4i)/Product_{j=1..i}(1 - x^j). - _Jon Perry_, Jul 04 2004
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (12*sqrt(2)*n^(5/2)). - _Vaclav Kotesovec_, Jun 02 2018
%F G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Aug 21 2018
%p series(1/product((1-x^i),i=4..65),x,60); # end of program
%p ZL := [ B,{B=Set(Set(Z, card>=4))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..60); # _Zerinvary Lajos_, Mar 13 2007
%t f[1, 1]=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, 4], {n, 60}] (* end of program *)
%t Drop[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*Length[p]]], {n, 60}],2] (* _Clark Kimberling_, Feb 27 2014 *)
%t Table[Count[IntegerPartitions[n],
%t p_ /; 3 Count[p, 1] == 2 Length[p]], {n, 0, 60}] (* _George Beck_ Aug 19 2017 *)
%t CoefficientList[Series[1/QPochhammer[x^4, x], {x,0,60}], x] (* _G. C. Greubel_, Nov 03 2019 *)
%o (Magma) a:= func< n | NumberOfPartitions(n)-NumberOfPartitions(n-1)-NumberOfPartitions(n-2)+ NumberOfPartitions(n-4)+NumberOfPartitions(n-5)- NumberOfPartitions(n-6) >; [1,0,0,0,1,1,1] cat [ a(n) : n in [7..60]]; // _Vincenzo Librandi_, Aug 20 2017
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/(&*[1-x^(m+4): m in [0..70]]) )); // _G. C. Greubel_, Nov 03 2019
%o (PARI) my(x='x+O('x^60)); Vec(1/prod(m=0,70, 1-x^(m+4))) \\ _G. C. Greubel_, Nov 03 2019
%o (Sage)
%o def A008484_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/product((1-x^(m+4)) for m in (0..70)) ).list()
%o A008484_list(60) # _G. C. Greubel_, Nov 03 2019
%Y 2-regular graphs with girth at least 4: A185114 (connected), A185224 (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), this sequence (g=4), A185325(g=5), A185326 (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).
%Y Not necessarily connected k-regular simple graphs with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: this sequence (k=2), A185334 (k=3), A185344 (k=4), A185354 (k=5), A185364 (k=6).
%K nonn,easy
%O 0,9
%A T. Forbes (anthony.d.forbes(AT)googlemail.com)
|
