%I #93 Feb 27 2024 10:43:28
%S 0,0,0,0,0,0,1,1,2,3,5,7,11,14,20,26,35,44,58,71,90,110,136,163,199,
%T 235,282,331,391,454,532,612,709,811,931,1057,1206,1360,1540,1729,
%U 1945,2172,2432,2702,3009,3331,3692,4070,4494,4935,5427,5942,6510,7104,7760
%N Number of partitions of n in which the greatest part is 6.
%C Also number of partitions of n into 6 parts. - _Washington Bomfim_, Jan 15 2021
%H Seiichi Manyama, <a href="/A026812/b026812.txt">Table of n, a(n) for n = 0..10000</a> (terms 1..1000 from Vincenzo Librandi)
%H G. E. Andrews, <a href="https://doi.org/10.1023/A:1026224002193">Partitions: At the Interface of q-Series and Modular Forms</a>, The Ramanujan Journal 7, 385-400 (2003), Eq.(3.10).
%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).
%F G.f.: x^6 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)). - _Colin Barker_, Dec 20 2012
%F a(n) = A008284(n,6). - _Robert A. Russell_, May 13 2018
%F a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} 1. - _Wesley Ivan Hurt_, Jun 29 2019
%F a(n) = A001402(n) - A001401(n). a(n) = A001402(n-6). - _Washington Bomfim_, Jan 15 2021
%F a(n) = round((1/86400)*n^5 + (1/3840)*n^4 + (19/12960)*n^3 - (n mod 2)*(1/384)*n^2 + (1/17280)*b(n mod 6)*n), where b(0)=96, b(1)=b(5)=-629, b(2)=b(4)=-224, and b(3)=-309. - _Washington Bomfim_ and _Jon E. Schoenfield_, Jan 16 2021
%t Table[ Length[ Select[ Partitions[n], First[ # ] == 6 & ]], {n, 1, 60} ]
%t CoefficientList[Series[x^6/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Oct 18 2013 *)
%t Drop[LinearRecurrence[{1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1}, Append[Table[0,{20}],1],115],14] (* _Robert A. Russell_, May 17 2018 *)
%o (PARI) my(x='x+O('x^99)); concat(vector(6), Vec(x^6/prod(k=1, 6, 1-x^k))) \\ _Altug Alkan_, May 17 2018
%o (PARI) a = vector(60,n,n--; round((n+11)*((6*n^4+249*n^3+2071*n^2 -4931*n+40621) /518400 +n\2*(n+10)/192+((n+1)\3+n\3*2)/54))); a = concat([0,0,0,0,0,0], a) \\ _Washington Bomfim_, Jan 16 2021
%o (GAP) List([0..70],n->NrPartitions(n,6)); # _Muniru A Asiru_, May 17 2018
%Y Essentially same as A001402.
%Y Cf. A026810, A026811, A026813, A026814, A026815, A026816.
%Y Cf. A001401, A001402.
%K nonn,easy
%O 0,9
%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