A026798 Number of partitions of n in which the least part is 5.

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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

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

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
• Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

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

Programs

Magma

R:=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

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 *)

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

Sage

def A026798_list(prec):
    P. = 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

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