A026814 Number of partitions of n in which the greatest part is 8.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1801, 2104, 2462, 2857, 3319, 3828, 4417, 5066, 5812, 6630, 7564, 8588, 9749, 11018, 12450, 14012, 15765, 17674

Offset

0,11

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, -1, 0, 1, 2, 1, 0, 1, -1, -1, -2, -1, -1, 1, 0, 1, 2, 1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, -1).

Formula

G.f.: x^8 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)). [Colin Barker, Feb 22 2013]

a(n) = A008284(n,8). - Robert A. Russell, May 13 2018

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} 1. - Wesley Ivan Hurt, Jul 04 2019

Mathematica

CoefficientList[Series[x^8/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) (1 - x^7) (1 - x^8)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Table[Count[IntegerPartitions[n], _?(Max[#]==8&)], {n, 0, 55}] (* Harvey P. Dale, Dec 04 2022 *)

Prog

(PARI) x='x+O('x^99); concat(vector(8), Vec(x^8/prod(k=1, 8, 1-x^k))) \\ Altug Alkan, May 17 2018
(GAP) List([0..70], n->NrPartitions(n, 8)); # Muniru A Asiru, May 17 2018

Crossrefs

Cf. A026810, A026811, A026812, A026813, A026815, A026816.

Sequence in context: A319474 A218508 A340719 * A008637 A008631 A347574

Adjacent sequences: A026811 A026812 A026813 * A026815 A026816 A026817

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

Two inoperative Mathematica programs deleted by Harvey P. Dale, Dec 04 2022

Status

approved