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A117485 Expansion of x^9/((1-x)*(1-x^2)*(1-x^3))^2. 18
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 (list; graph; refs; listen; history; text; internal format)
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: A025223 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

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Last modified September 23 09:03 EDT 2020. Contains 337298 sequences. (Running on oeis4.)