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Expansion of x^9/((1-x)*(1-x^2)*(1-x^3))^2.
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%I #31 Sep 08 2022 08:45:24

%S 1,2,5,10,18,30,49,74,110,158,221,302,407,536,698,896,1136,1424,1770,

%T 2176,2656,3216,3866,4616,5481,6466,7591,8866,10306,11926,13747,15778,

%U 18046,20566,23359,26446,29855,33600,37716,42224,47152,52528,58388,64752,71664

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

%C Molien series for S_3 X S_3, cf. A001399.

%C From _Gus Wiseman_, Apr 06 2019: (Start)

%C 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:

%C (333) (433) (443) (444) (544)

%C (3331) (533) (543) (553)

%C (3332) (633) (643)

%C (4331) (3333) (733)

%C (33311) (4332) (4333)

%C (4431) (4432)

%C (5331) (4441)

%C (33321) (5332)

%C (43311) (5431)

%C (333111) (6331)

%C (33322)

%C (33331)

%C (43321)

%C (44311)

%C (53311)

%C (333211)

%C (433111)

%C (3331111)

%C (End)

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

%F 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

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

%p 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

%t CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3))^2,{x,0,50}],x] (* _Harvey P. Dale_, Oct 09 2011 *)

%t durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];

%t Table[Length[Select[IntegerPartitions[n],durf[#]==3&]],{n,0,30}] (* _Gus Wiseman_, Apr 06 2019 *)

%o (Magma) n:=3; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // _N. J. A. Sloane_, Mar 10 2007

%o (PARI) Vec(x^9 / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^60)) \\ _Colin Barker_, Dec 12 2019

%Y Column k=3 of A115994.

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

%Y Cf. A115720, A307386, A325164.

%K nonn,easy

%O 9,2

%A _Alford Arnold_, Mar 22 2006

%E Entry revised by _N. J. A. Sloane_, Mar 10 2007