A268402 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly four times.

1, 7, 40, 204, 977, 4493, 20091, 88025, 379766, 1618898, 6835636, 28640302, 119236085, 493772409, 2035611612, 8359873866, 34219553297, 139672169795, 568675783762, 2310315996126, 9367885987455, 37920179012135, 153263612914150, 618611076034828, 2493830719572639, 10042451847789161

Offset: 5

Author

Ran Pan, Feb 03 2016

Keywords

• nonn

Comments

This sequence is related to paired pattern P_2 in Pan and Remmel's link.

By symmetry, it is also the number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the left exactly four times.

Links

• Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Crossrefs

Cf. A268400, A268401.

Formula

G.f.: -((-1 + f(x))^5*x^3*(-1 + f(x) + 2*x))/(2*(1 - f(x) + (-5 + f(x))*x)^5), where f(x) = sqrt(1 - 4*x).

Conjecture: -(n-3)*(n-5)*(55*n-618)*a(n) +(-55*n^3-840*n^2+6969*n-5742)*a(n-1)

+(3135*n^3 -37532*n^2 +138815*n -163614)*a(n-2) +(-7645*n^3 +93072*n^2 -343985*n +391386)*a(n-3) -18*(n-3)*(55*n-126)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Oct 07 2016