cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A268401 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly three times.

Original entry on oeis.org

1, 6, 31, 146, 655, 2850, 12148, 51028, 212039, 873870, 3578404, 14578576, 59149381, 239177006, 964439355, 3879856458, 15577609711, 62439489346, 249917575553, 999080441830, 3989726087031, 15917901326330, 63457122234055, 252796365476922, 1006456231070795, 4004838954237250, 15928299669867291
Offset: 4

Views

Author

Ran Pan, Feb 03 2016

Keywords

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 three times.

Links

Crossrefs

Cf. A268400.

Formula

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

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.

Original entry on oeis.org

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

Views

Author

Ran Pan, Feb 03 2016

Keywords

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

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

A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2*k,2*k).

Original entry on oeis.org

1, 2, 6, 14, 52, 140, 558, 1598, 6604, 19588, 82780, 251212, 1077992, 3324760, 14427422, 45039422, 197122524, 621205076, 2737289748, 8691699524, 38510822360, 123045322024, 547682980716, 1759017606220, 7859796084984, 25355507376808, 113670929821304
Offset: 0

Views

Author

John Tyler Rascoe, Oct 08 2024

Keywords

Examples

			The path NENNEENE does not cross y = x, so it is counted under a(4) = 52.
The path NENNENNEEEEN crosses y = x at points (1,1) and (5,5), so it is counted under a(6) = 558.

Crossrefs

Cf. A000108, A000984, A024492, A048990, A268400, A268429.

Programs

  • PARI
    C(x) = {(1-sqrt(1-4*x))/(2*x)}
A(x) ={C(4*x)*C((x)*C(4*x))}
B(x) = {sqrt(C(4*x))}
D(x) = {1/sqrt(1-4*x)}
E_x(N) = {my(x='x+O('x^N));  Vec(D(x)-2*((C(x)-1)*((x*A(x^2))^2-B(x^2)^2+3*B(x^2)-2))/((2-B(x^2))*(2-C(x))))}
E_x(30)

Formula

G.f. D(x) - 2*((C(x) - 1)*((x*A(x^2))^2 - B(x^2)^2 + 3*B(x^2) - 2))/((2 - B(x^2))*(2 - C(x))), where A(x), B(x), C(x), and D(x) are the g.f.s for A024492, A048990, A000108, and A000984.
Showing 1-3 of 3 results.

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