A294527 Number of Dyck paths of length 2n with exactly 2 hills.

0, 0, 1, 0, 3, 6, 21, 66, 220, 744, 2562, 8942, 31569, 112530, 404445, 1464042, 5332872, 19532688, 71893470, 265778040, 986416614, 3674092044, 13729259586, 51455182260, 193369903608, 728504292576, 2750904025276, 10409856537786, 39470613237645, 149935171349546

Offset: 0

Author

Eric M. Schmidt, Nov 01 2017

Keywords

• nonn

Links

• Dun Qiu and Jeffrey B. Remmel, Quadrant marked mesh patterns in 123-avoiding permutations, arXiv:1705.00164 [math.CO], 2017, p. 28.

Crossrefs

Column k=2 of A065600. Cf. A000957, A065601.

Programs

Mathematica

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];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 27 2018 *)

Formula

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