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A380453 Number of dessins d'enfants D(n,g) with n edges of genus g, read by rows.

%I A380453 #46 Jun 28 2025 11:50:22
%S A380453 1,3,6,1,20,6,60,33,4,291,285,48,1310,2115,708,30,6975,16533,9807,
%T A380453 1155,37746,126501,119436,29910,900,215602,972441,1355400,601364,
%U A380453 58032,1262874,7451679,14561360,10260804,2112300,54990,7611156,57167260,150429819,156469887,57017238,4764654
%N A380453 Number of dessins d'enfants D(n,g) with n edges of genus g, read by rows.
%C A380453 Note that Sum_{g>=0} D(n,g) gives A057005 which is the number of dessins d'enfants with n edges (as one would hope).
%C A380453 We get a new genus every two edges.
%C A380453 n=7 is the first time we have more dessins of genus 1 than genus 0.
%H A380453 Paawan Jethva, <a href="https://srs.amsi.org.au/wp-content/uploads/sites/92/2023/03/jethva_paawan_vrs-report.pdf">Exploring the Euler Characteristics of Dessins d’Enfants</a>, 2023, page 15.
%H A380453 Ján Karabáš, <a href="https://www.savbb.sk/~karabas/science.html#cyclic">Enumeration of actions of cyclic groups on orientable surfaces</a>.
%H A380453 A. Mednykh and R. Nedela, <a href="https://doi.org/10.1007/s10958-017-3555-5">Recent progress in enumeration of hypermaps</a>, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), tables 2-8 and theorems 8-12. Theorem 12 has typos; the corrected formula can be inferred from Karabáš's tables.
%e A380453 Triangle D(n,g) begins:
%e A380453    n\g    0      1      2      3      4      ...
%e A380453    1      1
%e A380453    2      3
%e A380453    3      6      1
%e A380453    4      20     6
%e A380453    5      60     33     4
%e A380453    6      291    285    48
%e A380453    7      1310   2115   708    30
%e A380453    8      6975   16533  9807   1155
%e A380453    9      37746  126501 119436 29910  900
%e A380453    ...
%Y A380453 Cf. A057005.
%Y A380453 Columns: A090371, A118094, A214819, A214820, A356694. A321710 is the rooted version.
%K A380453 nonn,tabf
%O A380453 1,2
%A A380453 _Paawan Jethva_, Jun 22 2025
%E A380453 Rows 10-11 from _Andrei Zabolotskii_, Jun 28 2025