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

Author

Clark Kimberling

Keywords

• nonn

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.

Crossrefs

Cf. A001791, A162551 (bisections).

Programs

Mathematica

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

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

Extensions

Definition corrected by Herbert Kociemba, May 08 2004