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A361360 Number of nonequivalent noncrossing caterpillars with n edges up to rotation and relection.

%I A361360 #8 Mar 11 2023 11:30:40
%S A361360 1,1,1,3,7,28,104,448,1886,8212,35556,155124,675897,2950074,12872294,
%T A361360 56188904,245253691,1070581703,4673231521,20399699635,89048927767,
%U A361360 388718917440,1696845506274,7407120344070,32333775400516,141144364258374,616127577376396
%N A361360 Number of nonequivalent noncrossing caterpillars with n edges up to rotation and relection.
%C A361360 The number of all noncrossing caterpillars with n edges is given by A361356.
%H A361360 Andrew Howroyd, <a href="/A361360/b361360.txt">Table of n, a(n) for n = 0..500</a>
%H A361360 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (5,4,-34,7,63,-30,-46,31,13,-14,0,3,-1).
%F A361360 G.f.: (1 - 4*x - 8*x^2 + 28*x^3 + 15*x^4 - 55*x^5 - 2*x^6 + 46*x^7 - 11*x^8 - 19*x^9 + 10*x^10 + 2*x^11 - 2*x^12)/((1 - x)^2*(1 + x)^2*(1 - 5*x + 3*x^2 - x^3)*(1 - 5*x^2 + 3*x^4 - x^6)).
%F A361360 a(n) = 5*a(n-1) + 4*a(n-2) - 34*a(n-3) + 7*a(n-4) + 63*a(n-5) - 30*a(n-6) - 46*a(n-7) + 31*a(n-8) + 13*a(n-9) - 14*a(n-10) + 3*a(n-12) - a(n-13) for n >= 13.
%o A361360 (PARI)
%o A361360 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/2 + x*subst((3 + 2*x*(3-x)*f)/(1-x)^2, x, x^2)/4 + subst(1/(1-x) + x*f/(1-x), x, x^2)/2}
%o A361360 { Vec(G(x) + O(x^30)) }
%Y A361360 Cf. A296533 (noncrossing trees), A361356, A361358, A361359 (up to rotation only).
%K A361360 nonn,easy
%O A361360 0,4
%A A361360 _Andrew Howroyd_, Mar 10 2023