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 A143023 #17 Jul 26 2022 10:52:38
%S A143023 1,6,41,306,2422,19980,169941,1479786,13127114,118217268,1077955034,
%T A143023 9932655348,92342765868,865126386072,8159523358029,77411610053658,
%U A143023 738263424935170,7073522484902820,68056887469098990,657269559836605980
%N A143023 Sum of the root degrees over all non-crossing connected graphs on n nodes on a circle (by root we mean a distinguished node).
%C A143023 The number of non-crossing connected graphs on n nodes on a circle is given in A007297.
%H A143023 Andrew Howroyd, <a href="/A143023/b143023.txt">Table of n, a(n) for n = 2..200</a>
%H A143023 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of noncrossing configurations</a>, Discrete Math. 204 (1999), 203-229.
%F A143023 a(n) = (L(n-2, 3n-2, n+1) + L(n-3, 3n-3, n))/(n-1) where L(p,q,r)=[u^p](1+u)^q/(1-u)^r = Sum_{i=0..min(p,q)} binomial(q,i) * binomial(r+p-1-i, r-1).
%F A143023 G.f.: zG(1+G)/(1-G) where G=G(z) satisfies G(1-G)=z(1+G)^3.
%F A143023 a(n) = Sum_{k=1..n-1} k*A143022(n,k).
%F A143023 D-finite with recurrence 5*n*(n-1)*a(n) -18*(n-1)*(n-3)*a(n-1) +12*(-45*n^2+180*n-178)*a(n-2) +216*(3*n-10)*(3*n-11)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%e A143023 a(3)=6 because in the graphs (AB,BC,CA), (AB,AC), (AB,BC) and (AC,BC) the root, say A, has degrees 2, 2, 1 and 1, respectively.
%p A143023 eq:=G*(1-G)-z*(1+G)^3: G:=RootOf(eq,G): Gser:=series(z*G*(1+G)/(1-G),z=0,25): seq(coeff(Gser,z,n),n=2..21); # end
%p A143023 L:=proc(p,q,r) options operator, arrow: sum(binomial(q,i)*binomial(r+p-1-i,r-1),i=0..min(p,q)) end proc: a:=proc(n) options operator, arrow: (L(n-2,3*n-2, n+1)+L(n-3,3*n-3,n))/(n-1) end proc: seq(a(n),n=2..21);
%t A143023 L[p_, q_, r_] := Sum[ Binomial[q, i]*Binomial[r + p - 1 - i, r-1], {i, 0, Min[p, q]}]; t[n_, k_] := 2^(k-1)*(k*L[n - k - 1, 3*n - k - 4, n - 2] - L[n - k - 2, 3*n - k - 3, n-1])/(n-1); t[2, 1] = 1; a[n_] := Sum[ k*t[n, k], {k, 1, n-1}]; Table[a[n], {n, 2, 21}] (* _Jean-François Alcover_, Oct 05 2011, after Maple *)
%o A143023 (PARI) {my(n=25); my(g=serreverse(x*(1-x)/(1+x)^3 + O(x*x^n))); Vec(g*(1+g)/(1-g))} \\ _Andrew Howroyd_, Dec 22 2017
%Y A143023 Cf. A143022, A007297.
%K A143023 nonn
%O A143023 2,2
%A A143023 _Emeric Deutsch_, Jul 30 2008