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

1,5

Links

Table of n, a(n) for n=1..23.

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.

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 *)

Crossrefs

Cf. A090371, A118094, A214820, A057005, A380453.

Sequence in context: A098402 A333481 A003774 * A211198 A179235 A387973

Adjacent sequences: A214816 A214817 A214818 * A214820 A214821 A214822

Keyword

nonn

Author

N. J. A. Sloane, Aug 01 2012

Extensions

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

Status

approved