This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A118445 #24 Feb 17 2024 03:22:50
%S A118445 1,25,490,8820,152460,2576574,42942900,709171320,11636856660,
%T A118445 190068658780,3093732938296,50222937310000,813611584422000,
%U A118445 13158602740363500,212528020730913000,3428785401125396400,55266606794455402500,890117467077758188500
%N A118445 Number of tree-rooted maps of genus 1 with n edges: rooted maps on the torus with a distinguished spanning tree.
%C A118445 Tree-rooted planar maps are counted by A005568 and tree-rooted maps of (orientable) genus 2 by A118446. Typically, a(11) = 190068658780 = 2^2*5*7^2*11*13^2*17^2*19^2.
%H A118445 E. A. Bender, E. R. Canfield and R. W. Robinson, <a href="https://doi.org/10.1016/0097-3165(88)90002-7">The asymptotic number of tree-rooted maps on a surface</a>, J. Comb. Theory, Ser. A, 48, No. 2 (1988), 156-164.
%H A118445 T. R. S. Walsh and A. B. Lehman, <a href="https://doi.org/10.1016/0095-8956(72)90049-4">Counting rooted maps by genus. II</a>, J. Comb. Theory, Ser. B, 13, No. 2 (1972), 122-141 (pp. 137, 140).
%F A118445 a(n) = binomial(2n, 0) C(0) b(n) + binomial(2n, 2) C(1) b(n-1) + binomial(2n, 4) C(2) b(n-2) + ... + binomial(2n, 2n) C(n) b(0), where C(n) = A000108(n) - n-th Catalan number and b(n) = (2n-1)!/(6(n-2)! (n-1)!) = A002802(n-2) - the number of toroidal one-vertex maps with n edges for n >= 2 and b(0) = b(1) = 0.
%F A118445 O.g.f.: x^2 * hypergeom([5/2, 5/2], [4], 16*x). - _Mark van Hoeij_, Apr 06 2013
%F A118445 D-finite with recurrence -(n+1)*(n-2)*a(n) +4*((2*n-1)^2)*a(n-1)=0. - _R. J. Mathar_, Jul 27 2022
%F A118445 From _Vaclav Kotesovec_, Feb 17 2024: (Start)
%F A118445 a(n) = n*(n-1) * binomial(2*n,n)^2 / (24*(n+1)).
%F A118445 a(n) ~ 2^(4*n-3)/(3*Pi). (End)
%t A118445 HypergeometricPFQ[{5/2, 5/2}, {4}, 16x] + O[x]^18 // CoefficientList[#, x]& (* _Jean-François Alcover_, Aug 28 2019 *)
%t A118445 Table[n*(n-1) * Binomial[2*n,n]^2 / (24*(n+1)), {n, 2, 20}] (* _Vaclav Kotesovec_, Feb 17 2024 *)
%K A118445 nonn
%O A118445 2,2
%A A118445 _Valery A. Liskovets_, May 04 2006
%E A118445 Added more terms, _Joerg Arndt_, Apr 07 2013