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A164680 Expansion of x/((1-x)^3*(1-x^2)^3*(1-x^3)).

%I A164680 #30 Aug 03 2025 16:16:41
%S A164680 1,3,9,20,42,78,139,231,372,573,861,1254,1791,2499,3432,4629,6162,
%T A164680 8085,10492,13455,17094,21503,26832,33201,40795,49764,60333,72687,
%U A164680 87096,103785,123075,145236,170646,199626,232617,269997,312277,359898,413448,473438
%N A164680 Expansion of x/((1-x)^3*(1-x^2)^3*(1-x^3)).
%C A164680 Convolution of A006918 with A001399, or of A002625 with A059841 (A000035 if offsets are respected),
%C A164680 or of A038163 with A022003 or of A057524 with A027656 or of A014125 with the aerated version of A000217,
%C A164680 or of A002624 with A103221, or of A002623 with A008731, or of other combinations of splitting the signature -/3,3,1 into two components.
%C A164680 If we apply the enumeration of Molien series as described in A139672,
%C A164680 this is row 45=9*5 of a table of values related to Molien series, i.e., the
%C A164680 product of the sequence on row 9 (A006918) with the sequence on row 5 (A001399).
%C A164680 This is associated with the root system E6, and can be described using the additive function on the affine E6 diagram:
%C A164680       1
%C A164680       |
%C A164680       2
%C A164680       |
%C A164680 1--2--3--2--1
%H A164680 G. C. Greubel, <a href="/A164680/b164680.txt">Table of n, a(n) for n = 1..1000</a>
%H A164680 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A164680 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).
%F A164680 a(n) = round( -(-1)^n*(n+3)*(n+7)/256 +(6*n^6 +180*n^5 +2070*n^4 +11400*n^3 +30429*n^2 +34290*n +9785)/103680 ) - _R. J. Mathar_, Mar 19 2012
%e A164680 To calculate a(3), we consider the first three terms of A001399 = (1 1 2...)
%e A164680 and the first three terms of A006918 = (1 2 5 ...), to get the convolved a(3) = 1*5+1*2+2*1 = 9.
%p A164680 seq(coeff(series(x/((1-x)^3*(1-x^2)^3*(1-x^3)), x, n+1), x, n), n = 1..40); # _G. C. Greubel_, Jan 13 2020
%t A164680 Rest@CoefficientList[Series[x/((1-x)^3*(1-x^2)^3*(1-x^3)), {x,0,40}], x] (* _G. C. Greubel_, Jan 13 2020 *)
%t A164680 LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{1,3,9,20,42,78,139,231,372,573,861,1254},40] (* _Harvey P. Dale_, Aug 03 2025 *)
%o A164680 (Sage)
%o A164680 x=PowerSeriesRing(QQ, 'x', 40).gen()
%o A164680 1/((1-x)^3*(1-x^2)^3*(1-x^3))
%o A164680 (PARI) Vec(1/(1-x)^3/(1-x^2)^3/(1-x^3)+O(x^40)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A164680 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^3*(1-x^2)^3*(1-x^3)) )); // _G. C. Greubel_, Jan 13 2020
%Y A164680 Cf. A139672 (row 21).
%Y A164680 For G2, the corresponding sequence is A001399.
%Y A164680 For F4, the corresponding sequence is A115264.
%Y A164680 For E7, the corresponding sequence is A210068.
%Y A164680 For E8, the corresponding sequence is A045513.
%Y A164680 See A210634 for a closely related sequence.
%K A164680 nonn,easy
%O A164680 1,2
%A A164680 _Alford Arnold_, Aug 21 2009
%E A164680 Edited and extended by _R. J. Mathar_, Aug 22 2009
%E A164680 Corrected link to index entries - _R. J. Mathar_, Aug 26 2009