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%I #19 Sep 08 2022 08:44:50
%S 10,340,20860,1770940,169271260,16731772540,1668294277660,
%T 166707356798140,16667683919380060,1666692097982207740,
%U 166667302449546018460,16666682561238613761340,1666667064030965197232860,166666676600774129343618940,16666666915019353231241663260
%N Number of words of length 2n in the 10 transpositions of S[5] equivalent to the identity.
%H Vincenzo Librandi, <a href="/A029698/b029698.txt">Table of n, a(n) for n = 1..300</a>
%H Justin Meiners, <a href="https://scholarsarchive.byu.edu/etd/8947">Computing the Rank of Braids</a>, Master's Thesis, Brigham Young University (2021) 8947.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (129,-3000,10000).
%F a(n) = (25*4^n+16*25^n +100^n)/60.
%F a(n) = 129*a(n-1)-3000*a(n-2)+10000*a(3). - _Colin Barker_, May 28 2015
%F G.f.: -10*x*(700*x^2-95*x+1) / ((4*x-1)*(25*x-1)*(100*x-1)). - _Colin Barker_, May 28 2015
%t Rest@ CoefficientList[Series[-10 x (700 x^2 - 95 x + 1)/((4 x - 1)*(25 x - 1)*(100 x - 1)), {x, 0, 15}], x] (* _Michael De Vlieger_, Jul 06 2021 *)
%o (Magma) [(25*4^n+16*25^n +100^n)/60: n in [1..15]]; // _Vincenzo Librandi_, Jun 30 2011
%o (PARI) Vec(-10*x*(700*x^2-95*x+1) / ((4*x-1)*(25*x-1)*(100*x-1)) + O(x^100)) \\ _Colin Barker_, May 28 2015
%K nonn,easy
%O 1,1
%A Paolo Dominici (pl.dm(AT)libero.it)
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