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A214820 Number of genus 3 sensed hypermaps with n darts.

%I A214820 #21 Jun 24 2025 14:03:11
%S A214820 0,0,0,0,0,0,30,1155,29910,601364,10260804,156469887,2195431068,
%T A214820 28897471080,361514582340,4339280187364,50323775391144,
%U A214820 566914469842923,6229721664499224,67000302262906866,707159710965012834,7341038807584085816,75093327553430134548
%N A214820 Number of genus 3 sensed hypermaps with n darts.
%H A214820 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 5.
%H A214820 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 A214820 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 A214820 hO[d_, _, _] := 0 /; !IntegerQ@d;
%t A214820 hO[d_, g_, q_] := Multinomial[d+2-2g-Total@q, Sequence@@q] h[g][d];
%t A214820 h[0][m_] := 3 2^(m-1) Binomial[2m, m] / ((m+1)(m+2));
%t A214820 h[1][d_] := Sum[2^k (4^(d-2-k)-1) Binomial[d+k, k], {k, 0, d-3}] / 3;
%t A214820 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 A214820 h[3][d_] := Coefficient[# (# - 1)^7(5 #^9 - 60 #^8 + 675 #^7 - 2947 #^6 + 10005 #^5 - 20235 #^4 + 28297 #^3 - 23937 #^2 + 11418 # - 1781)/(2 (# - 2)^12(# + 1)^9) &[(1-Sqrt[1-8x])/(4x) + O[x]^(d+1)],x,d];
%t A214820 a3[d_] := (h[3][d] + 15h[2][d/2] + 4hO[d/2,1,{4}] + hO[d/2,0,{8}] + 18hO[d/3,1,{2}] + 10hO[d/3,0,{5}] + 12hO[d/4,1,{2}] + 2hO[d/4,0,{3,2}] + 8hO[d/4,0,{4}] + 2hO[d/6,0,{1,2,1}] + 2hO[d/6,0,{2,2}] + 30hO[d/7,0,{3}] + 8hO[d/8,0,{1,2}] + 12hO[d/9,0,{1,2}] + 4hO[d/12,0,{1,2}] + 4hO[d/12,0,{1,1,1}] + 6hO[d/14,0,{1,1,1}]) / d;
%t A214820 Table[a3[n],{n,23}] (* _Andrei Zabolotskii_, Jun 24 2025, using Mednykh & Nedela's Theorem 9 *)
%Y A214820 Cf. A090371, A118094, A214819, A057005, A380453.
%K A214820 nonn
%O A214820 1,7
%A A214820 _N. J. A. Sloane_, Aug 01 2012
%E A214820 Terms a(13) onwards from _Andrei Zabolotskii_, Jun 24 2025