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

5,2

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

Table of n, a(n) for n=5..30.

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

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

Crossrefs

Cf. A268400, A268401.

Sequence in context: A055282 A252816 A093737 * A351529 A081039 A227748

Adjacent sequences: A268399 A268400 A268401 * A268403 A268404 A268405

Keyword

nonn

Author

Ran Pan, Feb 03 2016

Status

approved