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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.
2
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
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Definition corrected by Herbert Kociemba, May 08 2004
STATUS
approved

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Last modified September 20 15:18 EDT 2024. Contains 376072 sequences. (Running on oeis4.)