A026018 - OEIS

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.

%I #18 Jan 21 2026 12:08:46

%S 1,7,36,164,702,2898,11696,46512,183141,716243,2788060,10817820,

%T 41880930,161900910,625272480,2413491360,9313307370,35936613414,

%U 138680365704,535290282632,2066802226236,7983111461732,30848211650592,119257913003040,461268870161645

%N 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.

%H Michael De Vlieger, <a href="/A026018/b026018.txt">Table of n, a(n) for n = 3..1667</a>

%H Paul Drube, <a href="https://arxiv.org/abs/2206.01194">Raised k-Dyck paths</a>, arXiv:2206.01194 [math.CO], 2022. See Appendix pp. 14-15.

%F 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

%F From _Amiram Eldar_, Oct 12 2025: (Start)

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

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

%t a[n_] := Binomial[2*n-1, n-3] - Binomial[2*n-1, n-6]; Array[a, 30, 3] (* _Amiram Eldar_, Oct 12 2025 *)

%Y First differences of A003518.

%Y Cf. A026009.

%K nonn,easy

%O 3,2

%A _Clark Kimberling_