A214819 Number of genus 2 sensed hypermaps with n darts.
0, 0, 0, 0, 4, 48, 708, 9807, 119436, 1355400, 14561360, 150429819, 1506841872, 14732613116, 141226638540, 1331912032173, 12390368538412, 113927616087252, 1037080582036632, 9358430685657218, 83804192879934456, 745394788170961932, 6590038606472968276
Offset: 1
Keywords
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 4.
- 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.
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]; 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; Table[a2[n], {n, 23}] (* Andrei Zabolotskii, Jun 24 2025, using Mednykh & Nedela's Theorem 8 *)
Extensions
Terms a(13) onwards from Andrei Zabolotskii, Jun 24 2025
Comments