A026815 Number of partitions of n in which the greatest part is 9.

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, 252, 318, 393, 488, 598, 732, 887, 1076, 1291, 1549, 1845, 2194, 2592, 3060, 3589, 4206, 4904, 5708, 6615, 7657, 8824, 10156, 11648, 13338, 15224, 17354, 19720, 22380

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Essentially same as A008638.

Cf. A008284.

Cf. A026810, A026811, A026812, A026813, A026814, A026816.

Programs

GAP

List([0..70],n->NrPartitions(n,9)); # Muniru A Asiru, May 17 2018

Maple

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

Mathematica

Table[ Length[ Select[ Partitions[n], First[ # ] == 9 & ]], {n, 1, 60} ]
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 *)
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 *)

PARI

x='x+O('x^99); concat(vector(9), Vec(x^9/prod(k=1, 9, 1-x^k))) \\ Altug Alkan, May 17 2018

Formula

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

a(n) = A008284(n,9). - Robert A. Russell, May 13 2018

Extensions

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