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

0,10

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).

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

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 *)

Prog

(PARI) x='x+O('x^99); concat(vector(7), Vec(x^7/prod(k=1, 7, 1-x^k))) \\ Altug Alkan, May 17 2018
(GAP) List([0..70], n->NrPartitions(n, 7)); # Muniru A Asiru, May 17 2018
(Magma) [#Partitions(n, 7): n in [0..53]]; // Marius A. Burtea, Jul 01 2019

Crossrefs

Cf. A008284, A008636.

Cf. A026810, A026811, A026812, A026814, A026815, A026816.

Sequence in context: A309099 A218507 A339672 * A008636 A008630 A347573

Adjacent sequences: A026810 A026811 A026812 * A026814 A026815 A026816

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from Robert G. Wilson v, Jan 11 2002

a(0)=0 prepended by Seiichi Manyama, Jun 08 2017

Status

approved