A026018 a(n) = number of (s(0), s(1), ..., s(2*n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2*n-1) = 7. Also a(n) = T(2*n-1,n-3), where T is the array defined in A026009.

1, 7, 36, 164, 702, 2898, 11696, 46512, 183141, 716243, 2788060, 10817820, 41880930, 161900910, 625272480, 2413491360, 9313307370, 35936613414, 138680365704, 535290282632, 2066802226236, 7983111461732, 30848211650592, 119257913003040, 461268870161645

Offset

3,2

Links

Michael De Vlieger, Table of n, a(n) for n = 3..1667

Paul Drube, Raised k-Dyck paths, arXiv:2206.01194 [math.CO], 2022. See Appendix pp. 14-15.

Formula

Conjecture: -(n+5)*(3*n-37)*a(n) + 3*(-n^2-84*n-173)*a(n-1) + 2*(32*n^2+295*n+254)*a(n-2) - 8*(n+25)*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Jun 20 2013

From Amiram Eldar, Oct 12 2025: (Start)

a(n) = binomial(2*n-1, n-3) - binomial(2*n-1, n-6).

a(n) ~ 3 * 4^(n+1) / (n^(3/2) * sqrt(Pi)). (End)

Mathematica

a[n_] := Binomial[2*n-1, n-3] - Binomial[2*n-1, n-6]; Array[a, 30, 3] (* Amiram Eldar, Oct 12 2025 *)

Crossrefs

First differences of A003518.

Cf. A026009.

Sequence in context: A080420 A243037 A181292 * A085354 A051198 A003516

Adjacent sequences: A026015 A026016 A026017 * A026019 A026020 A026021

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved