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%I #39 Oct 29 2022 16:38:39
%S 1,1,1,2,2,3,4,5,6,8,10,12,15,17,21,25,29,34,40,46,53,62,70,80,91,103,
%T 116,131,147,164,184,204,227,252,278,307,339,372,408,448,489,534,583,
%U 634,689,749,811,878,950,1025,1106,1192,1282,1378,1481,1588,1702,1823,1949,2083
%N Expansion of 1/( Product_{j=0..5} (1-x^(2*j+1)) ).
%C Number of partitions of n into parts 1, 3, 5, 7, 9, and 11. - _Joerg Arndt_, Jul 09 2013
%C Number of partitions (d1,d2,...,d6) of n such that 0 <= d1/1 <= d2/2 <= ... <= d6/6. - _Seiichi Manyama_, Jun 04 2017
%H Seiichi Manyama, <a href="/A008675/b008675.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=246">Encyclopedia of Combinatorial Structures 246</a>
%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 2, -2, 3, -3, 3, -2, 3, -3, 3, -2, 2, -3, 2, -2, 2, -2, 1, -1, 1, -1, 1, 0, 1, -1).
%p seq(coeff(series(1/mul(1-x^(2*j+1), j=0..5), x, n+1), x, n), n = 0..65); # _G. C. Greubel_, Sep 08 2019
%t CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9)(1-x^11)), {x, 0, 65}], x] (* _Vincenzo Librandi_, Jun 23 2013 *)
%t LinearRecurrence[{1,0,1,-1,1,-1,1,-2,2,-2,2,-3,2,-2,3,-3,3,-2,3,-3,3,-2,2,-3,2,-2,2,-2,1,-1,1,-1,1,0,1,-1},{1,1,1,2,2,3,4,5,6,8,10,12,15,17,21,25,29,34,40,46,53,62,70,80,91,103,116,131,147,164,184,204,227,252,278,307},60] (* _Harvey P. Dale_, Oct 29 2022 *)
%o (PARI) a(n)=(46200*((n\3+1)*[2,-1,-1][n%3+1]+[10,-4,-7][n%3+1]) +3*n^5+ 270*n^4+9005*n^3+136350*n^2+908260*n+3603600)\3742200 \\ _Tani Akinari_, Jul 09 2013
%o (PARI) Vec(1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^9)*(1-x^11))+O(x^66)) \\ _Joerg Arndt_, Jul 09 2013
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/&*[1-x^(2*j+1): j in [0..5]] )); // _G. C. Greubel_, Sep 08 2019
%o (Sage)
%o def A008674_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/prod(1-x^(2*j+1) for j in (0..5)) ).list()
%o A008674_list(65) # _G. C. Greubel_, Sep 08 2019
%Y Cf. A259094.
%K nonn
%O 0,4
%A _N. J. A. Sloane_
%E Typo in name fixed by _Vincenzo Librandi_, Jun 23 2013
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