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Number of marked chord diagrams (linear words in which each letter appears twice) with n chords, whose intersection graph is connected and distance-hereditary.
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%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