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The 3rd Witt transform of A000027.
1

%I #18 Oct 25 2022 20:07:20

%S 0,0,0,0,2,7,18,42,84,153,264,429,666,1001,1456,2061,2856,3876,5166,

%T 6783,8778,11214,14168,17710,21924,26910,32760,39582,47502,56637,

%U 67122,79112,92752,108207,125664,145299,167310,191919,219336,249795,283556

%N The 3rd Witt transform of A000027.

%C a(n) is the number of binary Lyndon words of length n+3 having 3 blocks of 0's, see Math.SE. - _Andrey Zabolotskiy_, Nov 16 2021

%H Vincenzo Librandi, <a href="/A147611/b147611.txt">Table of n, a(n) for n = 0..1000</a>

%H Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.

%H Pieter Moree, <a href="http://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO].

%H Felix Pahl, <a href="https://math.stackexchange.com/a/3641671">Find the number of n-length Lyndon words on alphabet {0,1} with k blocks of 0's. (answer)</a>, Mathematics StackExchange, 2020.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,6,-9,12,-9,6,-6,4,-1).

%F G.f.: x^4*(2-x+2*x^2)/((1-x)^6*(1+x+x^2)^2).

%F a(n) = (1/27)*((3*A049347(n) + A049347(n-1)) - 3*(-1)^n*(A099254(n) - A099254(n- 1)) + n*(3*n^4 - 15*n^2 - 28)/40). - _G. C. Greubel_, Oct 24 2022

%t CoefficientList[Series[x^4(2 -x+ 2*x^2)/((1-x)^6*(1 +x +x^2)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 13 2012 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0,0,0,0] cat Coefficients(R!( x^4*(2-x+2*x^2)/((1-x)^6*(1+x+x^2)^2) )); // _G. C. Greubel_, Oct 24 2022

%o (SageMath)

%o def A147611_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( x^4*(2-x+2*x^2)/((1-x)^6*(1+x+x^2)^2) ).list()

%o A147611_list(50) # _G. C. Greubel_, Oct 24 2022

%Y Cf. A006584 (2nd Witt transform of A000027), A049347, A099254, A147618.

%K easy,nonn

%O 0,5

%A _R. J. Mathar_, Nov 08 2008