%I #70 Feb 29 2024 14:57:10
%S 0,0,0,0,0,0,0,1,1,2,3,5,7,11,15,21,28,38,49,65,82,105,131,164,201,
%T 248,300,364,436,522,618,733,860,1009,1175,1367,1579,1824,2093,2400,
%U 2738,3120,3539,4011,4526,5102,5731,6430,7190,8033,8946,9953,11044,12241
%N Number of partitions of n in which the greatest part is 7.
%H Seiichi Manyama, <a href="/A026813/b026813.txt">Table of n, a(n) for n = 0..10000</a> (terms 1..1000 from Vincenzo Librandi)
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, -1, 0, -1, -1, 0, 1, 1, 2, 0, 0, 0, -2, -1, -1, 0, 1, 1, 0, 1, 0, 0, -1, -1, 1).
%F G.f.: x^7 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)). - _Colin Barker_, Feb 22 2013
%F a(n) = A008284(n,7). - _Robert A. Russell_, May 13 2018
%F a(n) = A008636(n-7). - _R. J. Mathar_, Feb 13 2019
%F a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3} Sum_{i=j..floor((n-j-k-l-m-o)/2)} 1. - _Wesley Ivan Hurt_, Jun 30 2019
%t Table[ Length[ Select[ Partitions[n], First[ # ] == 7 & ]], {n, 1, 60} ]
%t CoefficientList[Series[x^7/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) (1 - x^7)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Oct 18 2013 *)
%t Drop[LinearRecurrence[{1,1,0,0,-1,0,-1,-1,0,1,1,2,0,0,0,-2,-1,-1,0,1,1,0,1,0,0,-1,-1,1}, Append[Table[0,{27}],1],121],20] (* _Robert A. Russell_, May 17 2018 *)
%o (PARI) x='x+O('x^99); concat(vector(7), Vec(x^7/prod(k=1, 7, 1-x^k))) \\ _Altug Alkan_, May 17 2018
%o (GAP) List([0..70],n->NrPartitions(n,7)); # _Muniru A Asiru_, May 17 2018
%o (Magma) [#Partitions(n,7): n in [0..53]]; // _Marius A. Burtea_, Jul 01 2019
%Y Cf. A008284, A008636.
%Y Cf. A026810, A026811, A026812, A026814, A026815, A026816.
%K nonn,easy
%O 0,10
%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