%I #21 Oct 08 2022 14:16:54
%S 1,4,27,226,2116,21218,222851,2420134,26954622,306203536,3534170486,
%T 41326973520,488562349730,5829471835390,70112478797987,
%U 849110215237094,10345827793291654,126734013316914248,1559884942820510474,19281814963272771308,239263099541276559360,2979328903819471935332
%N Number of marked chord diagrams (linear words in which each letter appears twice) with n chords, whose intersection graph is connected and distance-hereditary.
%C For n < 5, all intersection graphs on n vertices are distance-hereditary, so the first 4 terms coincide with the number of linear chord diagrams with connected intersection graph.
%H Christopher-Lloyd Simon, <a href="https://arxiv.org/abs/2106.15450">Topologie et dénombrement des courbes algébriques réelles</a>, arXiv:2106.15450 [math.AG], 2021.
%H Christopher-Lloyd Simon, <a href="https://doi.org/10.5802/afst.1698">Topologie et dénombrement des courbes algébriques réelles</a>, Annales de la faculté des sciences de Toulouse : Mathématiques, 6e série, 31(2): 383--422, 2022.
%F a(n) = (1/(n+1))*Sum_{k=0..n} binomial(n+k, n)*binomial(2*(n+1)+k, n-k)*2^k.
%F G.f. satisfies C = z + 2*z*C + (z+2)*C^2 + 2*C^3.
%o (PARI) a(n) = sum(k=0, n, binomial(n+k, n)*binomial(2*(n+1)+k, n-k)*2^k)/(n+1); \\ _Michel Marcus_, Oct 05 2022
%Y Cf. A277862, A277869, A357596.
%K nonn
%O 0,2
%A _Christopher-Lloyd Simon_, May 31 2022