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A026800 Number of partitions of n in which the least part is 7.

%I A026800 #49 Apr 05 2025 16:44:07
%S A026800 0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,4,4,5,6,7,8,10,11,
%T A026800 13,15,18,20,24,27,32,36,42,48,56,63,73,83,96,108,125,141,162,183,209,
%U A026800 236,270,304,346,390,443,498,565,635,719,807,911,1022,1153,1291,1453,1628,1829,2045
%N A026800 Number of partitions of n in which the least part is 7.
%C A026800 From _Jason Kimberley_, Feb 03 2011: (Start)
%C A026800 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.
%C A026800 By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. (End)
%H A026800 Robert Israel, <a href="/A026800/b026800.txt">Table of n, a(n) for n = 0..10000</a>
%H A026800 Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_k-reg_girth_eq_g_index">Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g</a>
%F A026800 G.f.: x^7 * Product_{m>=7} 1/(1-x^m).
%F A026800 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
%F A026800 a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - _Vaclav Kotesovec_, Jun 02 2018
%F A026800 G.f.: Sum_{k>=1} x^(7*k) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 25 2020
%e A026800 a(0)=0 because there does not exist a least part of the empty partition.
%e A026800 The  a(7)=1 partition is 7.
%e A026800 The a(14)=1 partition is 7+7.
%e A026800 The a(15)=1 partition is 7+8.
%e A026800 .............................
%e A026800 The a(20)=1 partition is 7+13.
%e A026800 The a(21)=2 partitions are 7+7+7 and 7+14.
%p A026800 N:= 100: # for a(0)..a(N)
%p A026800 S:= series(x^7/mul(1-x^i,i=7..N-7),x,N+1):
%p A026800 seq(coeff(S,x,i),i=0..N); # _Robert Israel_, Jul 04 2019
%t A026800 CoefficientList[Series[x^7/QPochhammer[x^7, x], {x, 0, 75}], x] (* _G. C. Greubel_, Nov 03 2019 *)
%t A026800 Join[{0},Table[Count[IntegerPartitions[n],_?(#[[-1]]==7&)],{n,80}]] (* _Harvey P. Dale_, Apr 05 2025 *)
%o A026800 (Magma) p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
%o A026800 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
%o A026800 (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
%o A026800 (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
%o A026800 (Sage)
%o A026800 def A026800_list(prec):
%o A026800     P.<x> = PowerSeriesRing(ZZ, prec)
%o A026800     return P( x^7/product((1-x^(m+7)) for m in (0..80)) ).list()
%o A026800 A026800_list(75) # _G. C. Greubel_, Nov 03 2019
%Y A026800 Cf. A185327 (Mathematica code)
%Y A026800 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).
%Y A026800 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
%K A026800 nonn,easy
%O A026800 0,22
%A A026800 _Clark Kimberling_
%E A026800 More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001