A026023 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 3. Also a(n) = Sum{T(n,k), k = 0,1,...,[ (n+3)/2 ]}, where T is defined in A026022.

1, 2, 4, 8, 15, 30, 56, 112, 210, 420, 792, 1584, 3003, 6006, 11440, 22880, 43758, 87516, 167960, 335920, 646646, 1293292, 2496144, 4992288, 9657700, 19315400, 37442160, 74884320, 145422675, 290845350, 565722720, 1131445440, 2203961430, 4407922860

Offset

0,2

Comments

a(n)/2^n is the probability that a random walker starting at x=4 and jumping +-1 with equal probability at each time step is not adsorbed at the boundary x=0 at time n. - Robert M. Ziff, Nov 10 2014

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018, Adv. Appl. Math. 101 (2018), 232-265.

Formula

a(2n) = C(2n+2, n), a(2n+1) = 2*a(2n).

E.g.f.: dif(Bessel_I(1,2x)+2*Bessel_I(2,2x)+Bessel_I(3,2x),x). - Paul Barry, Jun 09 2007

O.g.f.: -1/2*(-1+4*x^2+(1-8*x^2+20*x^4-16*x^6)^(1/2))/x^4/(2*x-1). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

Conjecture: (n+4)*(n-1)*a(n) +(n-1)*(n+1)*a(n-1) -2*(n+1)*(2*n+1)*a(n-2) -4*(n-1)*(n+1)*a(n-3)=0. - R. J. Mathar, Sep 29 2012

Mathematica

Module[{r=Range[0, 20], b}, Riffle[b=Binomial[2r+2, r], 2b]] (* Paolo Xausa, Dec 14 2023 *)

Crossrefs

Cf. A001791, A162551 (bisections).

Sequence in context: A217777 A034338 A166861 * A077596 A091865 A065494

Adjacent sequences: A026020 A026021 A026022 * A026024 A026025 A026026

Keyword

nonn

Author

Clark Kimberling

Extensions

Definition corrected by Herbert Kociemba, May 08 2004

Status

approved