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

0,16

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

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

Adjacent sequences: A026795 A026796 A026797 * A026799 A026800 A026801

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved