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A210696 Triangulations of the disk, G_{1,n}.

%I A210696 #30 Aug 08 2025 05:00:10
%S A210696 1,2,5,16,48,164,559,1952,6872,24520,88006,318444,1158944,4241688,
%T A210696 15598973,57620596,213680472,795270644,2969483214,11121038100,
%U A210696 41763779054,157235683780,593355907790,2243975358216,8503404201874,32283434698908,122779218918272,467713035691608
%N A210696 Triangulations of the disk, G_{1,n}.
%C A210696 This corrects a typographical error in A005497(6).
%H A210696 Andrew Howroyd, <a href="/A210696/b210696.txt">Table of n, a(n) for n = 0..500</a>
%H A210696 Jean-François Alcover, <a href="/A210696/a210696.txt">Mathematica program</a>
%H A210696 William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
%H A210696 William G. Brown, <a href="/A002709/a002709.pdf">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
%p A210696 BrownE := proc(r,n,m)
%p A210696     local j,s,p ;
%p A210696     if r < 1 then
%p A210696         return 0 ;
%p A210696     elif r = 1 then
%p A210696         return A146305(n,m) ;
%p A210696     elif r = 2 then
%p A210696         j := n mod 2 ; s := floor(n/2) ;
%p A210696         if type(m,'even') then
%p A210696             return 0 ;
%p A210696         end if;
%p A210696         p := (m+1)/2 ;
%p A210696         if p > 0 and s >= 0 then
%p A210696             return 2*(2*p)!*(4*s+2*p+2*j-1)!/p!/(p-1)!/s!/(3*s+2*p+2*j)! ;
%p A210696         else
%p A210696             return 0 ;
%p A210696         end if;
%p A210696     elif r =3 and (n mod 3) =0 and (m mod 3) = 0 then
%p A210696         s := n/3 ; p := m/3 ;
%p A210696         if p >= 0 and s >= 0 then
%p A210696             return (2*p+1)!*(4*s+2*p)!/p!/p!/s!/(3*s+2*p+1)! ;
%p A210696         else
%p A210696             return 0 ;
%p A210696         end if;
%p A210696     elif r >= 3 then
%p A210696         if ((n-1) mod r) =0 and ((m+3) mod r) =0 then
%p A210696             s := (n-1)/r ; p := (m+3)/r-1 ;
%p A210696             if p>=0 and s>=0 then
%p A210696             return (2*p+2)!*(4*s+2*p+1)!/p!/(p+1)!/s!/(3*s+2*p+2)! ;
%p A210696             else
%p A210696                 return 0 ;
%p A210696             end if;
%p A210696         else
%p A210696             return 0 ;
%p A210696         end if;
%p A210696     else
%p A210696         return 0 ;
%p A210696     end if;
%p A210696 end proc:
%p A210696 BrownG := proc(n,m)
%p A210696     add( numtheory[phi](s)* BrownE(s,n,m), s = numtheory[divisors](m+3) ) ;
%p A210696     %/(m+3) ;
%p A210696 end proc:
%p A210696 A210696 := proc(n)
%p A210696     BrownG(1,n) ;
%p A210696 end proc:
%p A210696 seq(A210696(n),n=0..25) ;
%t A210696 (* See the link section. *)
%Y A210696 Row n=1 of A262586.
%K A210696 nonn
%O A210696 0,2
%A A210696 _R. J. Mathar_, Mar 30 2012
%E A210696 a(26) onwards from _Andrew Howroyd_, Nov 23 2024