A360426 Number of permutations of [2n] having exactly n alternating up/down runs where the first run is not a down run.

1, 1, 6, 118, 4788, 325446, 33264396, 4766383420, 911323052520, 224136553339270, 68929638550210620, 25914939202996628148, 11693626371194331008088, 6236691723226152102621084, 3881046492003600271067466744, 2786922888404654795314066258488, 2287283298159853722760705106305488

Offset: 0

Author

Alois P. Heinz, Feb 08 2023

Keywords

• nonn

Comments

Number of permutations of [2n] such that the differences have n runs with the same signs where the first run does not have negative signs.

Examples

			a(0) = 1: (), the empty permutation.
a(1) = 1: 12.
a(2) = 6: 1243, 1342, 1432, 2341, 2431, 3421.
a(3) = 118: 123546, 123645, 124356, ..., 564123, 564213, 564312.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..228

Crossrefs

Cf. A000111, A008970, A059427.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, 0, `if`(k=1, 1,
      k*b(n-1, k) + 2*b(n-1, k-1) + (n-k)*b(n-1, k-2)))
    end:
a:= n-> `if`(n=0, 1, b(2*n, n)):
seq(a(n), n=0..17);

Formula

a(n) = A008970(2n,n) = (1/2) * A059427(2n,n) for n>=1.

a(n) ~ c * d^n * n!^2 / n, where d = 3.421054620671187024940215794079585351303138828348... (same as for A291677 and A303159) and c = 0.23613698601500409294656476488227001191406... - Vaclav Kotesovec, Feb 18 2023