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A361359 Number of nonequivalent noncrossing caterpillars with n edges up to rotation.

%I A361359 #8 Mar 11 2023 11:30:43
%S A361359 1,1,1,4,11,49,196,868,3721,16306,70891,309739,1350831,5897934,
%T A361359 25740386,112368153,490489041,2141121271,9346382981,40799215354,
%U A361359 178097506051,777437032059,3393689486976,14814237183658,64667544141561,282288713218896,1232255125682671
%N A361359 Number of nonequivalent noncrossing caterpillars with n edges up to rotation.
%C A361359 The number of all noncrossing caterpillars with n edges is given by A361356.
%H A361359 Andrew Howroyd, <a href="/A361359/b361359.txt">Table of n, a(n) for n = 0..500</a>
%H A361359 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (6,-3,-26,36,-2,-18,6,5,-4,1).
%F A361359 G.f.: (1 - 5*x - 2*x^2 + 27*x^3 - 20*x^4 - 13*x^5 + 23*x^6 - 5*x^7 - 6*x^8 + 3*x^9)/((1 - x)*(1 - 5*x + 3*x^2 - x^3)*(1 - 5*x^2 + 3*x^4 - x^6)).
%F A361359 a(n) = 6*a(n-1) - 3*a(n-2) - 26*a(n-3) + 36*a(n-4) - 2*a(n-5) - 18*a(n-6) + 6*a(n-7) + 5*a(n-8) - 4*a(n-9) + a(n-10) for n >= 10.
%o A361359 (PARI)
%o A361359 G(x)={ my(f = x*(2 - x)/(1 - 5*x + 3*x^2 - x^3), g = 1 + x + x^2*(3 - 2*x + (4 - 3*x + x^2)*f + (1 + 2*x)*f^2)/(1 - x)^2); (intformal(g) - 3)/x + x*subst((1 + 2*x*f)/(1-x)^2, x, x^2)/2 }
%o A361359 { Vec(G(x) + O(x^30)) }
%Y A361359 Cf. A296532 (noncrossing trees), A361356, A361358, A361360 (up to rotation and reflection).
%K A361359 nonn,easy
%O A361359 0,4
%A A361359 _Andrew Howroyd_, Mar 09 2023