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A206625 Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) in powers of x.

%I A206625 #18 Sep 08 2022 08:46:01
%S A206625 0,1,1,1,2,5,5,13,16,37,45,109,130,313,377,905,1088,2617,3145,7561,
%T A206625 9090,21853,26269,63157,75920,182525,219413,527509,634114,1524529,
%U A206625 1832625,4405969,5296384,12733489,15306833,36800465,44237570,106355317
%N A206625 Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4  - 2*x^6 + x^8) in powers of x.
%C A206625 This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).
%D A206625 J. A. Sjogren, Cycles and spanning trees. Math. Comput. Modelling 15, No.9, 87-102 (1991).
%H A206625 G. C. Greubel, <a href="/A206625/b206625.txt">Table of n, a(n) for n = 0..2500</a>
%H A206625 J. A. Sjogren, <a href="http://jon-arny.com/httpdocs/PDF/Cycles.pdf">Cycles and Spanning Trees</a>, see equation (3.5)
%H A206625 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A206625 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,2,0,2,0,-1).
%F A206625 G.f.: x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8).
%F A206625 a(n) = a(-n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8) for all n in Z.
%F A206625 a(2*n + 5) = A071100(n).  a(2*n + 6) = A071101(n). a(n + 3) = A112835(n). a(2*n) = A138573(n).
%e A206625 G.f. = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 13*x^7 + 16*x^8 + 37*x^9 + ...
%t A206625 CoefficientList[Series[x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 12 2018 *)
%o A206625 (PARI) {a(n) = my(m = abs(n)); polcoeff( x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) + x * O(x^m), m)};
%o A206625 (PARI) {a(n) = my(m = abs(n), v); v = polroots( Pol([ 1, 2, 4, 2, 1])); sqrtint( round( prod( k=1, 4, v[k]^m - 1, 2^(m%2) / 20)))};
%o A206625 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ))); // _G. C. Greubel_, Aug 12 2018
%Y A206625 Cf. A071100 (bisection), A071101 (bisection), A112835, A138573.
%K A206625 nonn,easy
%O A206625 0,5
%A A206625 _Michael Somos_, Feb 10 2012