A026812 Number of partitions of n in which the greatest part is 6.

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, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760

Offset

0,9

Comments

Also number of partitions of n into 6 parts. - Washington Bomfim, Jan 15 2021

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)

G. E. Andrews, Partitions: At the Interface of q-Series and Modular Forms, The Ramanujan Journal 7, 385-400 (2003), Eq.(3.10).

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Formula

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

a(n) = A008284(n,6). - Robert A. Russell, May 13 2018

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

a(n) = A001402(n) - A001401(n). a(n) = A001402(n-6). - Washington Bomfim, Jan 15 2021

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

Mathematica

Table[ Length[ Select[ Partitions[n], First[ # ] == 6 & ]], {n, 1, 60} ]
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 *)
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 *)

Prog

(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
(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
(GAP) List([0..70], n->NrPartitions(n, 6)); # Muniru A Asiru, May 17 2018

Crossrefs

Essentially same as A001402.

Cf. A026810, A026811, A026813, A026814, A026815, A026816.

Cf. A001401, A001402.

Sequence in context: A238659 A341870 A234666 * A001402 A008629 A347572

Adjacent sequences: A026809 A026810 A026811 * A026813 A026814 A026815

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