A214820 Number of genus 3 sensed hypermaps with n darts.

0, 0, 0, 0, 0, 0, 30, 1155, 29910, 601364, 10260804, 156469887, 2195431068, 28897471080, 361514582340, 4339280187364, 50323775391144, 566914469842923, 6229721664499224, 67000302262906866, 707159710965012834, 7341038807584085816, 75093327553430134548

Offset: 1

Author

N. J. A. Sloane, Aug 01 2012

Keywords

• nonn

Links

• A. Mednykh and R. Nedela, Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 5.
• Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
• Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.

Crossrefs

Cf. A090371, A118094, A214819, A057005, A380453.

Programs

Mathematica

hO[d_, , ] := 0 /; !IntegerQ@d;
hO[d_, g_, q_] := Multinomial[d+2-2g-Total@q, Sequence@@q] h[g][d];
h[0][m_] := 3 2^(m-1) Binomial[2m, m] / ((m+1)(m+2));
h[1][d_] := Sum[2^k (4^(d-2-k)-1) Binomial[d+k, k], {k, 0, d-3}] / 3;
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];
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];
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;
Table[a3[n],{n,23}] (* Andrei Zabolotskii, Jun 24 2025, using Mednykh & Nedela's Theorem 9 *)

Extensions

Terms a(13) onwards from Andrei Zabolotskii, Jun 24 2025