A091498 The sixth column of triangle A091492, excluding leading zeros.

1, 2, 3, 5, 8, 11, 17, 23, 31, 41, 54, 68, 88, 109, 135, 165, 202, 241, 291, 344, 407, 477, 559, 646, 751, 862, 990, 1129, 1288, 1456, 1651, 1857, 2089, 2338, 2617, 2911, 3244, 3594, 3982, 4395, 4851, 5330, 5862, 6420, 7031, 7677, 8382, 9120, 9929, 10775

Offset: 0

Author

Paul D. Hanna, Jan 16 2004

Keywords

• nonn

Comments

Excluding leading zeros, columns k=3,4,5, of triangle A091492 list the partitions of n into k parts.

This sequence is related to the partitions of n into at most 6 parts (A001402) since A(x)=(1+x-x^5)*G001402(x), where G001402(x) is the g.f. for A001402.

Links

• 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).

Crossrefs

Cf. A091492, A091493, A001402.

Programs

Mathematica

CoefficientList[Series[(1+x-x^5)/((1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)(1-x^6)),{x,0,60}],x] (* or *) LinearRecurrence[ {1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1},{1,2,3,5,8,11,17,23,31,41,54,68,88,109,135,165,202,241,291,344,407},60](* Harvey P. Dale, Dec 09 2012 *)

PARI

{a(n)=polcoeff( (1+x-x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)) +O(x^(n+1)),n,x)}

Formula

G.f.: (1+x-x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6))

a(0)=1, a(1)=2, a(2)=3, a(3)=5, a(4)=8, a(5)=11, a(6)=17, a(7)=23, a(8)=31, a(9)=41, a(10)=54, a(11)=68, a(12)=88, a(13)=109, a(14)=135, a(15)=165, a(16)=202, a(17)=241, a(18)=291, a(19)=344, a(20)=407, a(n)=a(n-1)+ a(n-2)- a(n-5)-2*a(n-7)+a(n-9)+a(n-10)+a(n-11)+a(n-12)-2*a(n-14)-a(n-16)+ a(n-19)+ a(n-20)-a (n-21). - Harvey P. Dale, Dec 09 2012