|
%I #14 Jul 26 2022 14:31:19
%S 1,3,13,66,367,2164,13293,84157,545270,3598244,24100375,163416748,
%T 1119592052,7738412716,53894849437,377851866954,2664570714233,
%U 18887568709788,134501247038230,961774761388430,6903072966224220
%N Number of noncrossing connected graphs on n nodes on a circle having no triangular faces.
%H Andrew Howroyd, <a href="/A045743/b045743.txt">Table of n, a(n) for n = 2..200</a>
%F a(n) = Sum_{i=0..floor((n-2)/2)} binomial(n-2+i, i)*binomial(3*n-3-i, n-2-2*i)/(n-1).
%F From _Paul D. Hanna_, Mar 09 2010: (Start)
%F G.f. A(x): Let F(x) = 1 + A(x)/x = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 +...
%F then F(x) satisfies: x*F(x)^4 = (1 - F(x))*(1 - 3*F(x) + F(x)^2). (End)
%F Conjecture D-finite with recurrence 5*n*(n-1)*(n-2)*(1353818*n-3651663)*a(n) +16*(n-1)*(n-2)*(2236406*n^2-6864025*n+2014500)*a(n-1) -16*(n-2)*(46338048*n^3-326486432*n^2+734160494*n-506411475)*a(n-2) +8*(2*n-7)*(26592864*n^3-436441232*n^2+2071128458*n-3030735075)*a(n-3) +128*(94246*n-334675)*(4*n-19)*(2*n-9)*(4*n-17)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1-x*A^4/(1-3*A+A^2));polcoeff(A,n)} \\ _Paul D. Hanna_, Mar 09 2010
%o (PARI) a(n) = if(n>1, sum(i=0, floor(n/2)-1, binomial(n-2+i, i)*binomial(3*n-3-i, n-2-2*i))/(n-1)); \\ _Andrew Howroyd_, Nov 12 2017
%Y Column k=0 of A089435.
%Y Cf. A045744.
%K nonn
%O 2,2
%A _Emeric Deutsch_
|
