A026802 Number of partitions of n in which the least part is 9.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 24, 26, 30, 34, 39, 43, 50, 55, 63, 71, 80, 89, 102, 113, 128, 143, 161, 179, 203, 225, 253, 282, 316, 351, 395, 437, 489, 544, 607, 673, 752, 832, 927, 1028, 1143

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Crossrefs

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), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), this sequence (g=9), A026803 (g=10).

Programs

Magma

R:=PowerSeriesRing(Integers(), 80); [0,0,0,0,0,0,0,0] cat Coefficients(R!( x^9/(&*[1-x^(m+9): m in [0..85]]) )); // G. C. Greubel, Nov 03 2019

Maple

seq(coeff(series(x^9/mul(1-x^(m+9), m = 0..85), x, n+1), x, n), n = 1..80); # G. C. Greubel, Nov 03 2019

Mathematica

Table[Count[IntegerPartitions[n],?(Min[#]==9&)],{n,80}] (* _Harvey P. Dale, May 09 2013 *)
Rest@CoefficientList[Series[x^9/QPochhammer[x^9, x], {x,0,80}], x] (* G. C. Greubel, Nov 03 2019 *)

PARI

my(x='x+O('x^70)); concat(vector(8), Vec(x^9/prod(m=0,85, 1-x^(m+9)))) \\ G. C. Greubel, Nov 03 2019

Sage

def A026802_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^9/product((1-x^(m+9)) for m in (0..85)) ).list()
a=A026802_list(81); a[1:] # G. C. Greubel, Nov 03 2019

Formula

G.f.: x^9 * Product_{m>=9} 1/(1-x^m).

a(n+9) = p(n) -p(n-1) -p(n-2) +p(n-5) +p(n-7) +p(n-9) -p(n-11) -2*p(n-12) -p(n-13) -p(n-15) +p(n-16) +p(n-17) +2*p(n-18) +p(n-19) +p(n-20) -p(n-21) -p(n-23) -2*p(n-24) -p(n-25) +p(n-27) +p(n-29) +p(n-31) -p(n-34) -p(n-35) +p(n-36) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010

a(n) ~ exp(Pi*sqrt(2*n/3)) * 70*Pi^8 / (9*sqrt(3)*n^5). - Vaclav Kotesovec, Jun 02 2018

G.f.: Sum_{k>=1} x^(9*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

Extensions

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001