A279208 Number of analytic chord diagrams with n chords, up to symmetry.
1, 2, 5, 17, 76, 499, 4132
Offset: 1
Links
- Étienne Ghys, A Singular Mathematical Promenade, arXiv:1612.06373 [math.AG], 2016. See p. 252.
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.
(* derived from Joerg Arndt's PARI code *) f[n_] := f[n] = (2n-1)!! s[n_] := s[n] = f[n] - Sum[f[k] s[n - k], {k, 1, n - 1}] Table[f[k] - s[k], {k, 0, 22}] (* original brute force method *) GenerateDiagramsOfOrder[n_Integer /; n >= 0] := Diagrams[Range[2 n]] Diagrams[pool_List] := Block[{n = Count[pool, _]}, If[n <= 2, {{pool}}, Flatten[Map[ Flatten[ Outer[Join, {{{pool[[1]], pool[[#]]}}}, Diagrams[ Function[{poolset, droppos}, Drop[poolset, {droppos}] // Rest][pool, #]], 1], 1] &, Range[2, n]], 1]]] SelectDisconnected[diagrams_List] := Select[diagrams, IsDisconnected] IsDisconnected[{{}}] = False; IsDisconnected[pairs_List] := Block[{newPairs=Map[#~Append~(#[[2]] - #[[1]]) &, pairs], distanceList}, distanceList = Fold[ ReplacePart[#1, {#2[[1]] -> #2[[3]], #2[[2]] -> -#2[[3]]}] &, Range[2 Length[pairs]], newPairs]; Return[Length[Select[Drop[Accumulate[distanceList], -1], #<1 &]] > 0] ] Map[Length[SelectDisconnected[GenerateDiagramsOfOrder[#]]]&, Range[0,7]]
f(n)=(2*n)!/n!/2^n; \\ == (2n-1)!! s(n)=f(n) - sum(k=1, n-1, f(k)*s(n-k) ) a(n)=f(n)-s(n) \\ Joerg Arndt
from functools import cache
def a(n):
@cache
def h(n):
if n <= 1: return 1
return h(n - 1) * (2 * n - 1)
@cache
def c(n):
if n == 0: return 1
return h(n) - sum(h(k) * c(n - k) for k in range(1, n))
return h(n) - c(n)
print([a(n) for n in range(19)]) # Peter Luschny, Apr 16 2023
Stoimenow states that a Mma package is available from his website. - N. J. A. Sloane, Jul 26 2018
Table[Floor[(2n-1)!!/(2n)],{n,20}] (* Harvey P. Dale, Nov 12 2012 *)
{ a357442(n) = ( sumdiv(n,d,(n\d)!*d^(n\d)*eulerphi(d)) + n*sum(k=0,n\2,n!\k!\2^k\(n-2*k)!) + if(n%2, n*((n-1)\2)!*2^((n-1)\2) + sumdiv(n,d, eulerphi(d)*sum(k=0,n\d\2,(n\d)! \ (2*k+1)! \ ((n\d-1)\2-k)! * (d/2)^((n\d-1)\2-k) ))) )\n\4; } \\ Max Alekseyev, Nov 10 2022
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