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%I #9 Jul 27 2018 12:37:25
%S 0,0,1,0,3,6,21,66,220,744,2562,8942,31569,112530,404445,1464042,
%T 5332872,19532688,71893470,265778040,986416614,3674092044,13729259586,
%U 51455182260,193369903608,728504292576,2750904025276,10409856537786,39470613237645,149935171349546
%N Number of Dyck paths of length 2n with exactly 2 hills.
%H Dun Qiu and Jeffrey B. Remmel, <a href="https://arxiv.org/abs/1705.00164">Quadrant marked mesh patterns in 123-avoiding permutations</a>, arXiv:1705.00164 [math.CO], 2017, p. 28.
%F Conjecture: 2*(3*n-5) *(n-2) *(3*n+11) *(n+1) *a(n) -(3*n+11) *(n-3) *(21*n^2-35*n+10) *a(n-1) -2*(3*n+11) *(n-1) *(2*n-1) *(3*n-2) *a(n-2)= 0. - _R. J. Mathar_, Jun 24 2018
%t a[n_] := Which[n>2, Sum[(i Binomial[i+2, i] Binomial[2n-2i-4, n-2])/(n-i-2), {i, 0, (n-2)/2}], n == 2, 1, True, 0];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 27 2018 *)
%Y Column k=2 of A065600. Cf. A000957, A065601.
%K nonn
%O 0,5
%A _Eric M. Schmidt_, Nov 01 2017