A026814 Number of partitions of n in which the greatest part is 8.
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1801, 2104, 2462, 2857, 3319, 3828, 4417, 5066, 5812, 6630, 7564, 8588, 9749, 11018, 12450, 14012, 15765, 17674
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
- Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, -1, 0, 1, 2, 1, 0, 1, -1, -1, -2, -1, -1, 1, 0, 1, 2, 1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, -1).
Programs
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GAP
List([0..70],n->NrPartitions(n,8)); # Muniru A Asiru, May 17 2018
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Mathematica
CoefficientList[Series[x^8/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) (1 - x^7) (1 - x^8)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *) Table[Count[IntegerPartitions[n],?(Max[#]==8&)],{n,0,55}] (* _Harvey P. Dale, Dec 04 2022 *) -
PARI
x='x+O('x^99); concat(vector(8), Vec(x^8/prod(k=1, 8, 1-x^k))) \\ Altug Alkan, May 17 2018
Formula
G.f.: x^8 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)). [Colin Barker, Feb 22 2013]
a(n) = A008284(n,8). - Robert A. Russell, May 13 2018
a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} 1. - Wesley Ivan Hurt, Jul 04 2019
Extensions
More terms from Robert G. Wilson v, Jan 11 2002
a(0)=0 prepended by Seiichi Manyama, Jun 08 2017
Two inoperative Mathematica programs deleted by Harvey P. Dale, Dec 04 2022