A026813 Number of partitions of n in which the greatest part is 7.
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, 248, 300, 364, 436, 522, 618, 733, 860, 1009, 1175, 1367, 1579, 1824, 2093, 2400, 2738, 3120, 3539, 4011, 4526, 5102, 5731, 6430, 7190, 8033, 8946, 9953, 11044, 12241
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, 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).
Programs
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GAP
List([0..70],n->NrPartitions(n,7)); # Muniru A Asiru, May 17 2018
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Magma
[#Partitions(n,7): n in [0..53]]; // Marius A. Burtea, Jul 01 2019
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Mathematica
Table[ Length[ Select[ Partitions[n], First[ # ] == 7 & ]], {n, 1, 60} ] 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 *) 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 *) -
PARI
my(x='x+O('x^99)); concat(vector(7), Vec(x^7/prod(k=1, 7, 1-x^k))) \\ Altug Alkan, May 17 2018
Formula
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
a(n) = A008284(n,7). - Robert A. Russell, May 13 2018
a(n) = A008636(n-7). - R. J. Mathar, Feb 13 2019
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
Extensions
More terms from Robert G. Wilson v, Jan 11 2002
a(0)=0 prepended by Seiichi Manyama, Jun 08 2017