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%I #49 Feb 27 2024 10:47:40
%S 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,
%T 288,352,434,525,638,764,919,1090,1297,1527,1801,2104,2462,2857,3319,
%U 3828,4417,5066,5812,6630,7564,8588,9749,11018,12450,14012,15765,17674
%N Number of partitions of n in which the greatest part is 8.
%H Seiichi Manyama, <a href="/A026814/b026814.txt">Table of n, a(n) for n = 0..10000</a> (terms 1..1000 from Vincenzo Librandi)
%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, 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).
%F 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]
%F a(n) = A008284(n,8). - _Robert A. Russell_, May 13 2018
%F 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
%t 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 *)
%t Table[Count[IntegerPartitions[n],_?(Max[#]==8&)],{n,0,55}] (* _Harvey P. Dale_, Dec 04 2022 *)
%o (PARI) x='x+O('x^99); concat(vector(8), Vec(x^8/prod(k=1, 8, 1-x^k))) \\ _Altug Alkan_, May 17 2018
%o (GAP) List([0..70],n->NrPartitions(n,8)); # _Muniru A Asiru_, May 17 2018
%Y Cf. A026810, A026811, A026812, A026813, A026815, A026816.
%K nonn,easy
%O 0,11
%A _Clark Kimberling_
%E More terms from _Robert G. Wilson v_, Jan 11 2002
%E a(0)=0 prepended by _Seiichi Manyama_, Jun 08 2017
%E Two inoperative Mathematica programs deleted by _Harvey P. Dale_, Dec 04 2022
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