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Expansion of 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^8)).
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%I #20 Jan 19 2021 20:07:39

%S 1,1,2,3,4,5,7,8,11,13,16,19,23,26,31,35,41,46,53,59,67,74,83,91,102,

%T 111,123,134,147,159,174,187,204,219,237,254,274,292,314,334,358,380,

%U 406,430,458,484,514,542,575,605,640,673,710,745,785,822,865,905

%N Expansion of 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^8)).

%C a(n) is equal to the number of partitions mu of n+6 of length 4 such that the transpose of mu has an even number of even entries (see below example). - _John M. Campbell_, Feb 02 2016

%C Number of partitions of n into parts 1, 2, 3, and 8. - _Michel Marcus_, Feb 03 2016

%C Number of partitions of n+4 into 4 parts whose smallest part is odd. - _Wesley Ivan Hurt_, Jan 19 2021

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1,0,1,-1,-1,0,1,1,-1).

%e From _John M. Campbell_, Feb 02 2016: (Start)

%e For example, letting n=6, there are a total of a(n)=a(6)=7 partitions mu of n+6=12 of length 4 such that the transpose of mu has an even number of even entries: (8,2,1,1), (6,4,1,1), (6,3,2,1), (6,2,2,2), (4,4,3,1), (4,4,2,2), (4,3,3,2). For example, the transpose of

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%e and contains 4 even entries. (End)

%t CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^8)), {x, 0, 200}], x] (* _John M. Campbell_, Feb 02 2016 *)

%o (PARI) Vec(1/((1-x)*(1-x^2)*(1-x^3)*(1-x^8)) + O(x^80)) \\ _Michel Marcus_, Feb 03 2016

%K nonn

%O 0,3

%A _N. J. A. Sloane_