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A289666 a(n) = number of weakly threshold graphs on n nodes.

%I A289666 #17 Sep 15 2024 11:45:19
%S A289666 0,1,2,4,9,21,52,134,355,957,2608,7154,19701,54379,150302,415762,
%T A289666 1150609,3185147,8818620,24418128,67615743,187239359,518506932,
%U A289666 1435875288,3976322869,11011542937,30494088494,84446895364,233857897749,647620493541
%N A289666 a(n) = number of weakly threshold graphs on n nodes.
%H A289666 Colin Barker, <a href="/A289666/b289666.txt">Table of n, a(n) for n = 0..1000</a>
%H A289666 Michael D. Barrus, <a href="https://arxiv.org/abs/1608.01358">Weakly threshold graphs</a>, arXiv preprint arXiv:1608.01358 [math.CO], 2016. See W(x).
%H A289666 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-1,0,-1).
%F A289666 G.f.: -(x^4+x^2+2*x-1)*x/((x^2+x-1)*(x^3-x^2+3*x-1)).
%F A289666 a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3) - a(n-5) for n>5. - _Colin Barker_, Jul 26 2017
%F A289666 3*a(n) = 2*A000045(n+1) + A200752(n+3) -2*A200752(n+2), n>0. - _R. J. Mathar_, Aug 05 2017
%t A289666 a = DifferenceRoot[Function[{a, n}, {a[n] + a[n+2] + 3a[n+3] - 4a[n+4] + a[n+5] == 0, a[0]==0, a[1]==1, a[2]==2, a[3]==4, a[4]==9, a[5]==21}]];
%t A289666 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 09 2019 *)
%t A289666 LinearRecurrence[{4,-3,-1,0,-1},{0,1,2,4,9,21},30] (* _Harvey P. Dale_, Sep 15 2024 *)
%o A289666 (PARI) concat(0, Vec(x*(1 - 2*x - x^2 - x^4) / ((1 - x - x^2)*(1 - 3*x + x^2 - x^3)) + O(x^30))) \\ _Colin Barker_, Jul 26 2017
%K A289666 nonn,easy
%O A289666 0,3
%A A289666 _N. J. A. Sloane_, Jul 25 2017