0,6

Also number of Dyck paths of semi-length n for which the number of valleys added to the number of triple falls is k.

Apparently deletion of zeros and row-reversal maps A166073 to A091156. - R. J. Mathar, Oct 08 2009

Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 03 2009: (Start)

The trivariate o.g.f. G=G(t,s,x), where t marks triple falls, s marks valleys, and x marks semilength is given by G=1+x[1+xg+t(G-1-xg)]g, where g = s(G-1)+1. Letting t=s=y, yields the given o.g.f.

(End)

M. Barnabei, F. Bonetti and M. Silimbani, The descent statistic on 123 avoiding permutations, preprint

O.g.f. E(x,y)=(-1+2xy+2x^2y-2xy^2-4x^2y^2+2x^2y^3+Sqrt[1-4xy-4x^2y+4*x^2*y^2])/(2xy^2(xy-1-x))

For example, for n=4 and k=1 whe have the 2 permutations 3412 and 2413. Triangle begins: 1; 1,1; 0,4,1; 0,2,11,1; 0,0,15,26,1; 0,0,5,59,57,1; 0,0,0,56,252,120,1; ...

G := (-1+2*x*y+2*x^2*y-2*x*y^2-4*x^2*y^2+2*x^2*y^3+sqrt(1-4*x*y-4*x^2*y+4*x^2*y^2))/(\ 2*x*y^2*(x*y-1-x)): Gser := simplify(series(G, x = 0, 17)): for n from 0 to 12 do P[n] := sort(expand(coeff(Gser, x, n))) end do: for n from 0 to 12 do seq(coeff(P[n], y, k), k = 0 .. n-1) end do; # yields sequence in triangular form [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 30 2009]

E[x_, y_]:=(-1+2*x*y+2*x^2*y-2*x*y^2-4*x^2*y^2+2*x^2*y^3+Sqrt[1-4*x*y-4*x^2*y+4*x^2*y^\ 2])/(2*x*y^2*(x*y-1-x))

Cf. A001263. Row sums given by A000108.

Cf. A091156. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2009]

Sequence in context: A096501 A062862 A084119 this_sequence A122899 A021713 A122388

Adjacent sequences: A166070 A166071 A166072 this_sequence A166074 A166075 A166076

nonn,new,tabf

Matteo Silimbani (silimban(AT)dm.unibo.it), Oct 06 2009, Oct 08 2009

Extended by Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 30 2009