A026122 a(n) is the number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n), where T is the array in A026120.

2, 4, 11, 28, 74, 196, 525, 1416, 3846, 10508, 28864, 79664, 220818, 614460, 1715874, 4807008, 13506534, 38052972, 107477319, 304261404, 863188662, 2453737132, 6988033949, 19935797080, 56966012730, 163026450132, 467219178549, 1340810339036

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Crossrefs

Cf. A001006, A026120.

First differences of A026107.

Programs

Mathematica

Drop[CoefficientList[Series[-1 + (1 - x)^2*(-1 + x + Sqrt[1 - 2*x - 3*x^2])^2 / (4*x^4), {x, 0, 30}], x], 2] (* Vaclav Kotesovec, Sep 17 2019 *)

Maxima

a(n):=2*sum((binomial(2*m+1,m)*binomial(n-1,2*m-1))/(m+2),m,1,n/2); /* Vladimir Kruchinin, Jan 24 2022 */

Formula

G.f.: (-1 + (1-z)^2 * M^2), with M the g.f. of the Motzkin numbers (A001006). [corrected by Vaclav Kotesovec, Sep 17 2019]

Conjecture: (n+4)*a(n) +(-3*n-5)*a(n-1) +(-n-6)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 23 2013

a(n) ~ 4 * 3^(n + 1/2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 17 2019

a(n) = 2*Sum_{m=1..n/2} C(2*m+1,m)*C(n-1,2*m-1)/(m+2). - Vladimir Kruchinin, Jan 24 2022