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A026799
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Number of partitions of n in which the least part is 6.
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20
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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
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OFFSET
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0,19
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COMMENTS
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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.
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LINKS
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FORMULA
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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
G.f.: Sum_{k>=1} x^(6*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
Join[{0}, Table[Count[IntegerPartitions[n][[;; , -1]], 6], {n, 70}]] (* Harvey P. Dale, Dec 27 2023 *)
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PROG
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(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)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^6/product((1-x^(m+6)) for m in (0..70)) ).list()
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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