A227656 Number of lattice paths from {2}^n to {0}^n using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_n) we have abs(p_{i}-p_{i+1}) <= 1.

1, 1, 4, 44, 896, 29392, 1413792, 93770800, 8201380224, 914570667792, 126651310675680, 21323599202141616, 4289517397262212416, 1016086393608958657680, 279937626985917460931616, 88754294249179769383418160, 32085579878185717054048193280, 13119328150439580260369558815248

Offset

0,3

Comments

Number of linear extensions of garland or double fence poset. - Alexander Shashkov, Jul 26 2020

Links

Alexander Shashkov, Table of n, a(n) for n = 0..227 (terms 0..23 from Alois P. Heinz)

Oscar J. Borenstein and Alexander Shashkov, Garland Recurrences, arXiv:1909.04215 [math.CO], 2019.

Jiaxi Lu and Yuanzhe Ding, A skeleton model to enumerate standard puzzle sequences, arXiv:2106.09471 [math.CO], 2021.

Formula

a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.197278552664313325820060688708960349... and c = 4.4668518532326348084863454883501... - Vaclav Kotesovec, Dec 25 2018

Example

a(2) = 2^2 = 4:
.
        (1,2)       (0,1)
       /     \     /     \
  (2,2)       (1,1)       (0,0)
       \     /     \     /
        (2,1)       (1,0)
.
a(3) = 44:
.
          (1,2,2)-(1,1,2)-(0,1,2)-(0,1,1)-(0,0,1)
         /       X       \       /       X       \
  (2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0)
         \       X       /       \       X       /
          (2,2,1)-(2,1,1)-(2,1,0)-(1,1,0)-(1,0,0)

Crossrefs

Row n=2 of A227655.

Cf. A000079.

Sequence in context: A370058 A144004 A240318 * A206686 A302909 A053333

Adjacent sequences: A227653 A227654 A227655 * A227657 A227658 A227659

Keyword

nonn

Author

Alois P. Heinz, Jul 19 2013

Status

approved