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%I #32 Sep 08 2022 08:45:15
%S 1,1,2,2,4,4,7,7,11,11,16,16,23,23,31,31,41,41,53,53,67,67,83,83,102,
%T 102,123,123,147,147,174,174,204,204,237,237,274,274,314,314,358,358,
%U 406,406,458,458,514,514,575,575,640,640,710,710,785,785,865,865,950,950,1041,1041
%N Expansion of 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)).
%C Molien series for symmetrized weight enumerators of Hermitian self-dual codes over the Galois ring GR(4,2).
%C Number of partitions of n into parts 1, 2, 4, and 6. - _Joerg Arndt_, May 05 2014
%C a(n) is equal to the number of partitions of degree at most n+6 of length 3 with even entries. - _John M. Campbell_, Jan 20 2016
%H G. C. Greubel, <a href="/A099770/b099770.txt">Table of n, a(n) for n = 0..1000</a>
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,0,0,-1,1,-1,1,1,-1).
%F a(n) ~ 1/288*n^3. - _Ralf Stephan_, Apr 29 2014
%F It appears that a(n) = (2*n^3 +39*n^2 +241*n +372 +3*(n^2 +13*n +40) * (-1)^n -84*(-1)^((2*n +3 +(-1)^n)/4) -192*floor(((2*n +9 +(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/24)))/576. - _Luce ETIENNE_, May 05 2014
%e From _John M. Campbell_, Jan 20 2016: (Start)
%e Letting n=6, a(n) = 7 is equal to the number of partitions of n into parts 1, 2, 4, and 6, as illustrated below, and a(n) is equal to the number of partitions of degree at most n+6 of length 3 with even entries, as illustrated below. The arrows below illustrate a natural bijection between the set of partitions of the former form and the set of partitions of the latter form.
%e (2, 2, 2) <-> (1, 1, 1, 1, 1, 1)
%e (4, 2, 2) <-> (2, 1, 1, 1, 1)
%e (6, 2, 2) <-> (4, 1, 1)
%e (4, 4, 2) <-> (2, 2, 1, 1)
%e (8, 2, 2) <-> (6)
%e (6, 4, 2) <-> (4, 2)
%e (4, 4, 4) <-> (2, 2, 2)
%e (End)
%p seq(coeff(series(1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)), x, n+1), x, n), n = 0 .. 65); # _G. C. Greubel_, Sep 04 2019
%t CoefficientList[Series[1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)), {x, 0, 65}], x] (* _G. C. Greubel_, Sep 04 2019 *)
%o (PARI) Vec(1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)) + O(x^80)) \\ _Michel Marcus_, Jan 21 2016
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)) )); // _G. C. Greubel_, Sep 04 2019
%o (Sage)
%o def A099770_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6))).list()
%o A099770_list(65) # _G. C. Greubel_, Sep 04 2019
%o (GAP) a:=[1,1,2,2,4,4,7,7,11,11,16,16,23];; for n in [14..65] do a[n]:= a[n-1]+a[n-2]-a[n-3]+a[n-4]-a[n-5]-a[n-8]+a[n-9]-a[n-10]+a[n-11]+a[n-12] -a[n-13]; od; a; # _G. C. Greubel_, Sep 04 2019
%Y Cf. A000601.
%K nonn
%O 0,3
%A G. Nebe (nebe(AT)math.rwth-aachen.de), Nov 10 2004
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