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A026798
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Number of partitions of n in which the least part is 5.
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21
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1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 13, 15, 18, 21, 26, 30, 36, 42, 50, 58, 70, 80, 95, 110, 129, 150, 176, 202, 236, 272, 317, 364, 423, 484, 560, 643, 740, 847, 975, 1112, 1277, 1456, 1666, 1897, 2168
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OFFSET
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0,16
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COMMENTS
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Also the number of not necessarily connected 2-regular simple graphs with girth exactly 5. - Jason Kimberley, Nov 11 2011
Such partitions of n+5 correspond to A185325 partitions (parts >= 5) of n by removing a single part of size 5. - Jason Kimberley, Nov 11 2011
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LINKS
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FORMULA
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G.f.: x^5 * Product_{m>=5} 1/(1-x^m).
a(n+5) is given by p(n) - p(n-1) - p(n-2) + 2p(n-5) - p(n-8) - p(n-9) + p(n-10) where p(n) = A000041(n). - Shanzhen Gao, Oct 28 2010 [sign of 10 and offset of formula corrected by Jason Kimberley, Nov 11 2011]
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (6*sqrt(3)*n^3). - Vaclav Kotesovec, Jun 02 2018
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MAPLE
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ZL := [ B, {B=Set(Set(Z, card>=5))}, unlabeled ]: 1, 0, 0, 0, 0, seq(combstruct[count](ZL, size=n), n=0..54); # Zerinvary Lajos, Mar 13 2007
1, seq(coeff(series(x^5/mul(1-x^(m+5), 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] = 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[{1, 0, 0, 0, 0, 1}, Table[ f[n, 5], {n, 50}]] (* Robert G. Wilson v *)
Join[{1}, Drop[CoefficientList[Series[x^5/QPochhammer[x^5, x], {x, 0, 60}], x], 1]] (* G. C. Greubel, Nov 03 2019 *)
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PROG
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(PARI) my(x='x+O('x^60)); concat([1, 0, 0, 0, 0], Vec(x^5/prod(m=0, 70, 1-x^(m+5)))) \\ G. C. Greubel, Nov 03 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); [1, 0, 0, 0, 0] cat Coefficients(R!( x^5/(&*[1-x^(m+5): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^5/product((1-x^(m+5)) 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), this sequence (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Nov 11 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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