A026017 - OEIS

A026017

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) = 5. Also a(n) = T(2*n-1,n-2), where T is the array defined in A026009.

1, 5, 21, 83, 319, 1209, 4550, 17068, 63954, 239666, 898909, 3375825, 12697035, 47833905, 180510210, 682341000, 2583591150, 9798281910, 37218303330, 141585223494, 539395269462, 2057771255210, 7860697923436, 30065829471048, 115135255095140, 441410428339972

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OFFSET

2,2

LINKS

Michael De Vlieger, Table of n, a(n) for n = 2..1667

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

FORMULA

Expansion of (1+x^1*C^3)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

Conjecture: (n+4)*a(n) + (-8*n-17)*a(n-1) + (19*n+1)*a(n-2) + 6*(-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-2) - binomial(2*n-1, n-5).

a(n) ~ 9 * 4^n / (n^(3/2) * sqrt(Pi)). (End)

MATHEMATICA

a[n_] := Binomial[2*n-1, n-2] - Binomial[2*n-1, n-5]; Array[a, 30, 2] (* Amiram Eldar, Oct 12 2025 *)

CROSSREFS

First differences of A003517.

Cf. A000108, A026009.

Sequence in context: A217783 A221862 A216271 * A132310 A083319 A146041

Adjacent sequences: A026014 A026015 A026016 * A026018 A026019 A026020

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved