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

1, 1, 1, 3, 7, 28, 104, 448, 1886, 8212, 35556, 155124, 675897, 2950074, 12872294, 56188904, 245253691, 1070581703, 4673231521, 20399699635, 89048927767, 388718917440, 1696845506274, 7407120344070, 32333775400516, 141144364258374, 616127577376396

Offset: 0

Author

Andrew Howroyd, Mar 10 2023

Keywords

• nonn
• easy

Comments

The number of all noncrossing caterpillars with n edges is given by A361356.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (5,4,-34,7,63,-30,-46,31,13,-14,0,3,-1).

Crossrefs

Cf. A296533 (noncrossing trees), A361356, A361358, A361359 (up to rotation only).

Programs

PARI

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}
{ Vec(G(x) + O(x^30)) }

Formula

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)).

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.