%I #53 Feb 29 2024 14:59:41
%S 0,0,0,0,0,0,0,0,0,1,1,2,3,5,7,11,15,22,30,41,54,73,94,123,157,201,
%T 252,318,393,488,598,732,887,1076,1291,1549,1845,2194,2592,3060,3589,
%U 4206,4904,5708,6615,7657,8824,10156,11648,13338,15224,17354,19720,22380
%N Number of partitions of n in which the greatest part is 9.
%H Seiichi Manyama, <a href="/A026815/b026815.txt">Table of n, a(n) for n = 0..10000</a> (terms 1..1000 from Vincenzo Librandi)
%H <a href="/index/Rec#order_45">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 2, 1, 1, 1, 0, -1, -1, -1, -2, -1, -1, 1, 1, 2, 1, 1, 1, 0, -1, -1, -1, -2, 0, 1, 0, 0, 1, 0, 1, 0, 0, -1, -1, 1).
%F G.f.: x^9 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)*(1-x^9)). - _Colin Barker_, Feb 22 2013
%F a(n) = A008284(n,9). - _Robert A. Russell_, May 13 2018
%p part_ZL:=[S,{S=Set(U,card=r),U=Sequence(Z,card>=1)}, unlabeled]: seq(count(subs(r=9,part_ZL),size=m),m=1..50); # _Zerinvary Lajos_, Mar 09 2007
%t Table[ Length[ Select[ Partitions[n], First[ # ] == 9 & ]], {n, 1, 60} ]
%t CoefficientList[Series[x^9/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) (1 - x^7) (1 - x^8) (1 - x^9)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Oct 18 2013 *)
%t Drop[LinearRecurrence[{1, 1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 2, 1, 1, 1, 0, -1, -1, -1, -2, -1, -1, 1, 1, 2, 1, 1, 1, 0, -1, -1, -1, -2, 0, 1, 0, 0, 1, 0, 1, 0, 0, -1, -1, 1}, Append[Table[0,{44}],1],136],35] (* _Robert A. Russell_, May 17 2018 *)
%o (PARI) x='x+O('x^99); concat(vector(9), Vec(x^9/prod(k=1, 9, 1-x^k))) \\ _Altug Alkan_, May 17 2018
%o (GAP) List([0..70],n->NrPartitions(n,9)); # _Muniru A Asiru_, May 17 2018
%Y Essentially same as A008638.
%Y Cf. A008284.
%Y Cf. A026810, A026811, A026812, A026813, A026814, A026816.
%K nonn,easy
%O 0,12
%A _Clark Kimberling_
%E a(0)=0 prepended by _Seiichi Manyama_, Jun 08 2017