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 A188442 #37 Jan 30 2020 22:07:55
%S A188442 0,1,1,3,6,16,39,103,269,721,1941,5285,14476,39918,110633,308081,
%T A188442 861390,2417264,6805477,19216297,54404562,154402264,439172189,
%U A188442 1251701837,3574234089,10223988665,29292665059,84052426683,241518890124,694900077226
%N A188442 Expansion of -(sqrt(-3*x^2-2*x+1)-x-1)/(2*sqrt(-3*x^2-2*x+1)+2*x).
%C A188442 Apparently for n>0 the number of grand Motzkin paths of length n-1 that avoid flat steps at level 1. - _David Scambler_, Jul 04 2013
%C A188442 a(n) is the number of increasing unary-binary trees with associated permutation that simultaneously avoids 213 and 312. For more information about increasing unary-binary trees with an associated permutation, see A245888. - _Manda Riehl_, Aug 22 2014
%F A188442 G.f.: A000045(x*A005043(x))
%F A188442 a(n)=sum(k=1..n,k*A000045(k)*sum(j=k..n, binomial(-k+2*j-1,j-1)*(-1)^(n-j)*binomial(n,j)))/n.
%F A188442 a(n)=sum(k=1..n, T097609(n,k)*A000045(k)).
%F A188442 D-finite with recurrence n*(5*n-2)*a(n) +(-20*n^2+23*n-12)*a(n-1) +3*(-5*n^2+7*n+20)*a(n-2) +2*(35*n^2-89*n-12)*a(n-3) +12*(5*n+3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jul 23 2017
%t A188442 a[n_] := Sum[ k*Fibonacci[k]*Sum[ Binomial[-k+2*j-1, j-1]*(-1)^(n-j)*Binomial[n, j], {j, k, n}], {k, 1, n}]/n; a[0] = 0; Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Jun 14 2013, translated from Maxima *)
%o A188442 (Maxima)
%o A188442 a(n):=sum(k*fib(k)*sum(binomial(-k+2*j-1,j-1)*(-1)^(n-j)*binomial(n,j),j,k,n),k,1,n)/n;
%o A188442 (PARI) x='x+O('x^66);
%o A188442 gf=-(sqrt(-3*x^2-2*x+1)-x-1)/(2*sqrt(-3*x^2-2*x+1)+2*x); /* = 0 +x +x^2 +3*x^3 +... */
%o A188442 Vec(gf) /* _Joerg Arndt_, Apr 21 2011 */
%Y A188442 A000984 is the sequence of the odd-indexed terms of a(n).
%K A188442 nonn
%O A188442 0,4
%A A188442 _Vladimir Kruchinin_, Apr 20 2011