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A214819 Number of genus 2 sensed hypermaps with n darts.

%I A214819 #24 Jun 24 2025 10:37:56
%S A214819 0,0,0,0,4,48,708,9807,119436,1355400,14561360,150429819,1506841872,
%T A214819 14732613116,141226638540,1331912032173,12390368538412,
%U A214819 113927616087252,1037080582036632,9358430685657218,83804192879934456,745394788170961932,6590038606472968276
%N A214819 Number of genus 2 sensed hypermaps with n darts.
%H A214819 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), table 4.
%H A214819 Timothy R. Walsh, <a href="http://www.info2.uqam.ca/~walsh_t/papers/GENERATING NONISOMORPHIC.pdf">Space-efficient generation of nonisomorphic maps and hypermaps</a>
%H A214819 Timothy R. Walsh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Walsh/walsh3.html">Space-Efficient Generation of Nonisomorphic Maps and Hypermaps</a>, J. Int. Seq. 18 (2015) # 15.4.3.
%t A214819 hO[d_, _, _] := 0 /; !IntegerQ@d;
%t A214819 hO[d_, g_, q_] := Multinomial[d+2-2g-Total@q, Sequence@@q] h[g][d];
%t A214819 h[0][m_] := 3 2^(m-1) Binomial[2m,m] / ((m+1)(m+2));
%t A214819 h[1][d_] := Sum[2^k (4^(d-2-k)-1) Binomial[d+k,k], {k,0,d-3}] / 3;
%t A214819 h[2][d_] := Coefficient[-# (# - 1)^5 (#^4 - 6 #^3 + 36 #^2 - 50 # + 51) / (4 (# - 2)^7 (# + 1)^5) &[(1-Sqrt[1-8x])/(4x)+O[x]^(d+1)], x, d];
%t A214819 a2[d_] := (h[2][d] + 4hO[d/2,1,{2}] + hO[d/2,0,{6}] + 6hO[d/3,0,{0,4}] + 2hO[d/4,0,{2,0,2}] + 12hO[d/5,0,{0,0,0,3}] + 2hO[d/6,0,{2,2}] + 2hO[d/6,0,{0,1,0,0,2}] + 4hO[d/8,0,{1,0,0,0,0,0,2}] + 4hO[d/10,0,{1,0,0,1,0,0,0,0,1}]) / d;
%t A214819 Table[a2[n], {n, 23}] (* _Andrei Zabolotskii_, Jun 24 2025, using Mednykh & Nedela's Theorem 8 *)
%Y A214819 Cf. A090371, A118094, A214820, A057005, A380453.
%K A214819 nonn
%O A214819 1,5
%A A214819 _N. J. A. Sloane_, Aug 01 2012
%E A214819 Terms a(13) onwards from _Andrei Zabolotskii_, Jun 24 2025