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A045743 Number of noncrossing connected graphs on n nodes on a circle having no triangular faces. 3

%I

%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)

%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_

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Last modified August 5 11:15 EDT 2021. Contains 346467 sequences. (Running on oeis4.)