A129115 - OEIS

%I #12 Jan 29 2025 22:11:20

%S 1,3,17,124,1618,33564,956263,33736198,1402665692,66902717187,

%T 3596481426812,215049652739982,14154852098315796,1016911004448831247,

%U 79174846391508487198,6640511488761139957873,596865894849670793348763,57234563024075319273338452,5832189914390355126473955563

%N Number of unrooted unlabeled not necessarily connected triangular maps on a compact closed oriented surface with 2n faces (and thus 3n edges).

%C Equivalently, the number of pair of permutations (sigma,tau) up to simultaneous conjugacy on a set of size 6*n with sigma^3=tau^2=1 with no fixed point.

%H Andrew Howroyd, <a href="/A129115/b129115.txt">Table of n, a(n) for n = 0..300</a>

%F Euler transform of A129114. - _Andrew Howroyd_, Jan 29 2025

%o (PARI)

%o D(m,k)={my(g=gcd(m,k)); sumdiv(g, d, my(j=m/d); x^j*eulerphi(d)*k^(j-1)/j)}

%o seq(n)={my(t=6*n); Vec(prod(k=1, t, my(A=O(x^(t\k+1)), p=serconvol(exp(A + D(3,k)), exp(A + D(2,k)))); sum(r=0, t\k, if(k*r%6==0, r!*polcoef(p,r)/(k^r)*x^(k*r/6)), O(x*x^n)) ))} \\ _Andrew Howroyd_, Jan 29 2025

%Y Not necessarily connected version of A129114.

%Y Unrooted, not necessarily connected version of A062980.

%Y Cf. also A121350, A121352, A005133.

%K nonn,changed

%O 0,2

%A _Samuel A. Vidal_, Mar 30 2007

%E a(17) onwards from _Andrew Howroyd_, Jan 28 2025