A268429 -id:A268429 - cp's OEIS frontend

A268430 Number of North-East paths from (0,0) to (n,n) that have even number of times bounce off y = x to the right.

1, 2, 5, 16, 53, 184, 654, 2368, 8689, 32216, 120434, 453248, 1715266, 6521584, 24894364, 95353472, 366324729, 1411015064, 5447548682, 21074836864, 81682204614, 317110750672, 1232951721604, 4800353915264, 18712880651802, 73030245143792, 285311648317172

Offset: 0

Ran Pan, Feb 04 2016

nonn

This sequence is related to paired pattern P_2 in Section 3.2 in Pan and Remmel's link.

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

Cf. A268407, A268429

G.f.: (-1 + 3*f(x) - 2*x*(-2 + 5*f(x)+ 6*f(x)*x))/(2*(4*x - 1) (-1 + 4*x*(1 + x))), where f(x) = sqrt(1 - 4*x).

a(n) = binomial(2*n,n) - A268431(n).

A268431 Number of North-East paths from (0,0) to (n,n) that have odd number of times bounce off y = x to the right.

Original entry on oeis.org

1, 4, 17, 68, 270, 1064, 4181, 16404, 64322, 252184, 988890, 3879016, 15222236, 59764048, 234755661, 922591156, 3627586618, 14270426936, 56164324206, 221147123768, 871147242116, 3433076812336, 13534723031298, 53380361293960, 210606884630932

Offset: 2

Ran Pan, Feb 04 2016

nonn

This sequence is related to paired pattern P_2 in Section 3.2 in Pan and Remmel's link.

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

Cf. A268407, A268429, A268430.

a(n) = binomial(2*n,n) - A268430(n).

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

A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 4, 5, 4, 4, 3, 2, 1, 1, 1, 4, 7, 10, 11, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 5, 11, 18, 24, 27, 30, 29, 28, 25, 23, 19, 16, 12, 10, 7, 5, 3, 2, 1, 1, 1, 6, 16, 30, 46, 59, 71, 78, 81, 81, 78, 74, 67, 60, 52, 46, 37, 31, 24

Offset: 0

John Tyler Rascoe, Aug 26 2024

Here the weight of a lattice path is the area under the path and above the x-axis. T(n,k) also counts the number of integer compositions of (3*n) + (2*k) + 6 with adjacent differences in {-1,1}, first part 1, and last part 3.

			Triangle begins:
    k=0  1  2   3   4   5   6   7   8   9  10  11  12  13  14
 n=0: 1;
 n=1: 1, 1, 1;
 n=2: 1, 2, 2,  2,  1,  1;
 n=3: 1, 3, 4,  5,  4,  4,  3,  2,  1,  1;
 n=4: 1, 4, 7, 10, 11, 11, 11,  9,  8,  6,  5,  3,  2,  1,  1;
 ...
T(1,0) = 1: (NENN).
T(2,1) = 2: (NNEENN) and (NENNEN).
T(3,2) = 4: (NENENNNE), (NENNENEN), (NNEENNEN), and (NNENEENN).

John Tyler Rascoe, Python program.

Cf. A000245 (empirical row sums), A000217 (row lengths).
Cf. A227543 (paths of this kind from (0,0) to (n,n), offset 1 for (0,0) to (n,n+1)).
Cf. A000108, A152659, A173258, A227543, A268429.

Python
```
# see linked program
```

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 (2k,2k).

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

John Tyler Rascoe, Oct 08 2024

			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.

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

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)

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.

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A268430 Number of North-East paths from (0,0) to (n,n) that have even number of times bounce off y = x to the right.

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A268431 Number of North-East paths from (0,0) to (n,n) that have odd number of times bounce off y = x to the right.

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A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

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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 (2k,2k).

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cp's OEIS Frontend

A268430 Number of North-East paths from (0,0) to (n,n) that have even number of times bounce off y = x to the right.

Views

Author

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A268431 Number of North-East paths from (0,0) to (n,n) that have odd number of times bounce off y = x to the right.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Python

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).

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Author

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PARI

Formula

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 (2k,2k).