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

9,2

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)

Links

Table of n, a(n) for n=9..53.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

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

Example

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

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 *)

Prog

(Magma) n:=3; G:=SymmetricGroup(n); H:=DirectProduct(G, G); MolienSeries(H); // N. J. A. Sloane, Mar 10 2007
(PARI) Vec(x^9 / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Dec 12 2019

Crossrefs

Column k=3 of A115994.

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

Cf. A115720, A307386, A325164.

Sequence in context: A348919 A177787 A104688 * A084835 A298107 A034350

Adjacent sequences: A117482 A117483 A117484 * A117486 A117487 A117488

Keyword

nonn,easy

Author

Alford Arnold, Mar 22 2006

Extensions

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

Status

approved