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A026798 Number of partitions of n in which the least part is 5. 21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,16
COMMENTS
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
LINKS
FORMULA
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(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)
def A026798_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^5/product((1-x^(m+5)) for m in (0..70)) ).list()
a=A026798_list(65); [1]+a[1:] # G. C. Greubel, Nov 03 2019
CROSSREFS
Essentially the same as A185325.
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).
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
Sequence in context: A096749 A036821 A237980 * A185325 A125890 A067661
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified April 16 23:10 EDT 2024. Contains 371755 sequences. (Running on oeis4.)