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

%I A117485 #55 Jul 17 2025 17:53:48
%S A117485 1,2,5,10,18,30,49,74,110,158,221,302,407,536,698,896,1136,1424,1770,
%T A117485 2176,2656,3216,3866,4616,5481,6466,7591,8866,10306,11926,13747,15778,
%U A117485 18046,20566,23359,26446,29855,33600,37716,42224,47152,52528,58388,64752,71664
%N A117485 Expansion of x^9/((1-x)*(1-x^2)*(1-x^3))^2.
%C A117485 Molien series for S_3 X S_3, cf. A001399.
%C A117485 From _Gus Wiseman_, Apr 06 2019: (Start)
%C A117485 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 A117485   (333)  (433)   (443)    (444)     (544)
%C A117485          (3331)  (533)    (543)     (553)
%C A117485                  (3332)   (633)     (643)
%C A117485                  (4331)   (3333)    (733)
%C A117485                  (33311)  (4332)    (4333)
%C A117485                           (4431)    (4432)
%C A117485                           (5331)    (4441)
%C A117485                           (33321)   (5332)
%C A117485                           (43311)   (5431)
%C A117485                           (333111)  (6331)
%C A117485                                     (33322)
%C A117485                                     (33331)
%C A117485                                     (43321)
%C A117485                                     (44311)
%C A117485                                     (53311)
%C A117485                                     (333211)
%C A117485                                     (433111)
%C A117485                                     (3331111)
%C A117485 (End)
%H A117485 <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).
%H A117485 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%F A117485 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
%F A117485 From _Hoang Xuan Thanh_, May 17 2025: (Start)
%F A117485 a(n+3) = Sum_{x+2*y+3*z=n} x*y*z.
%F A117485 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)
%e A117485 As a cross-check, row sixteen of A115994 yields p(16) = 16 + 140 + 74 + 1.
%p A117485 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 A117485 CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3))^2,{x,0,50}],x] (* _Harvey P. Dale_, Oct 09 2011 *)
%t A117485 durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
%t A117485 Table[Length[Select[IntegerPartitions[n],durf[#]==3&]],{n,0,30}] (* _Gus Wiseman_, Apr 06 2019 *)
%o A117485 (Magma) n:=3; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // _N. J. A. Sloane_, Mar 10 2007
%o A117485 (PARI) Vec(x^9 / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^60)) \\ _Colin Barker_, Dec 12 2019
%o A117485 (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
%Y A117485 Column k=3 of A115994.
%Y A117485 Cf. A000027 (for k=1), A006918 (for k=2), A117488, A117489, A001399, A117486.
%Y A117485 Cf. A115720, A307386, A325164.
%K A117485 nonn,easy
%O A117485 9,2
%A A117485 _Alford Arnold_, Mar 22 2006
%E A117485 Entry revised by _N. J. A. Sloane_, Mar 10 2007