cp's OEIS Frontend

A268558 Number of not necessarily connected sensed combinatorial maps with n edges.

%I A268558 #28 Jan 31 2025 11:39:55
%S A268558 1,2,8,34,182,1300,12634,153598,2231004,37250236,699699968,
%T A268558 14574247086,333121322514,8286605836248,222824153996898,
%U A268558 6439779836400464,199051769194393718,6552226226766384216,228826838199807593530,8450335361750379998822,329002470731473098130572
%N A268558 Number of not necessarily connected sensed combinatorial maps with n edges.
%C A268558 Original name: Arises in counting maps on a surface: see Coquereaux-Zuber (2015) for precise definition.
%C A268558 Number of nonisomorphic pairs (s,t) of permutations on a 2n-set where t is a fixed point free involution (i.e. all 2-cycles). Isomorphism is up to permutations of the n-set. - _Andrew Howroyd_, Jan 28 2025
%H A268558 Andrew Howroyd, <a href="/A268558/b268558.txt">Table of n, a(n) for n = 0..400</a> (terms 0..30 from Alois P. Heinz)
%H A268558 R. de Mello Koch, S. Ramgoolam, <a href="https://doi.org/10.1103/PhysRevD.85.026007">Strings from Feynman graph counting: Without large N</a>, Phys. Rev. D 85 (2012) 026007. Different from a(7) onwards.
%H A268558 R. Coquereaux, J.-B. Zuber, <a href="http://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, <a href="https://dx.doi.org/10.1142/S0218216516500474">1650047</a> (2016),
%o A268558 (PARI)
%o A268558 b(k,r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
%o A268558 S(n,k)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k,r)), O(x*x^n))}
%o A268558 seq(n)={Vec(prod(k=1, 2*n, S(n,k)))} \\ _Andrew Howroyd_, Jan 28 2025
%Y A268558 Euler transform of A170946.
%K A268558 nonn
%O A268558 0,2
%A A268558 _N. J. A. Sloane_, Mar 02 2016
%E A268558 a(11)-a(18) from Euler transform of A170946 - _R. J. Mathar_, Apr 07 2022
%E A268558 a(0)=1 prepended and a(19)-a(20) (via A170946) from _Alois P. Heinz_, Jan 27 2025
%E A268558 Name edited by _Andrew Howroyd_, Jan 31 2025