A117485 Expansion of x^9/((1-x)*(1-x^2)*(1-x^3))^2.

1, 2, 5, 10, 18, 30, 49, 74, 110, 158, 221, 302, 407, 536, 698, 896, 1136, 1424, 1770, 2176, 2656, 3216, 3866, 4616, 5481, 6466, 7591, 8866, 10306, 11926, 13747, 15778, 18046, 20566, 23359, 26446, 29855, 33600, 37716, 42224, 47152, 52528, 58388, 64752, 71664

Offset: 9

Author

Alford Arnold, Mar 22 2006

Keywords

• nonn
• easy

Comments

Molien series for S_3 X S_3, cf. A001399.

From Gus Wiseman, Apr 06 2019: (Start)

Also the number of integer partitions of n with Durfee square of length 3. The Heinz numbers of these partitions are given by A307386. For example, the a(9) = 1 through a(13) = 18 partitions are:

(333) (433) (443) (444) (544)

(3331) (533) (543) (553)

(3332) (633) (643)

(4331) (3333) (733)

(33311) (4332) (4333)

(4431) (4432)

(5331) (4441)

(33321) (5332)

(43311) (5431)

(333111) (6331)

(33322)

(33331)

(43321)

(44311)

(53311)

(333211)

(433111)

(3331111)

(End)

Examples

			As a cross-check, row sixteen of A115994 yields p(16) = 16 + 140 + 74 + 1.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).
• Index entries for two-way infinite sequences

Crossrefs

Column k=3 of A115994.

Cf. A000027 (for k=1), A006918 (for k=2), A117488, A117489, A001399, A117486.

Cf. A115720, A307386, A325164.

Programs

Magma

n:=3; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // N. J. A. Sloane, Mar 10 2007

Maple

with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=3, stack): seq(count(subs(r=3, ZL), size=m), m=6..50) ; # Zerinvary Lajos, Jan 02 2008

Mathematica

CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3))^2,{x,0,50}],x] (* Harvey P. Dale, Oct 09 2011 *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==3&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)

PARI

Vec(x^9 / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Dec 12 2019

PARI

a(n) = floor((3*n^5 - 45*n^4 + 200*n^3 - 180*n^2 - 363*n + 1600)/12960 + n/27*(n%3==0) - n/32*(n%2==0)) \\ Hoang Xuan Thanh, Jul 17 2025

Formula

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>20. - Colin Barker, Dec 12 2019

From Hoang Xuan Thanh, May 17 2025: (Start)

a(n+3) = Sum_{x+2*y+3*z=n} x*y*z.

a(n+3) = n*(n^2-1)*(3*n^2-67)/12960 - floor((n+1)/3)/27 + [n mod 2 = 0]*n/32 + [n mod 3 = 0]*n/27 where [] is the Iverson bracket. (End)

Extensions

Entry revised by N. J. A. Sloane, Mar 10 2007