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

2,1

Links

Table of n, a(n) for n=2..29.

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

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 *)

Prog

(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 */

Crossrefs

Cf. A001006, A026120.

First differences of A026107.

Sequence in context: A202085 A122423 A099016 * A108629 A007048 A148132

Adjacent sequences: A026119 A026120 A026121 * A026123 A026124 A026125

Keyword

nonn

Author

Clark Kimberling

Status

approved