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%I #42 Dec 14 2023 12:27:25
%S 1,2,4,8,15,30,56,112,210,420,792,1584,3003,6006,11440,22880,43758,
%T 87516,167960,335920,646646,1293292,2496144,4992288,9657700,19315400,
%U 37442160,74884320,145422675,290845350,565722720,1131445440,2203961430,4407922860
%N 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.
%C 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
%H Paolo Xausa, <a href="/A026023/b026023.txt">Table of n, a(n) for n = 0..1000</a>
%H Christian Krattenthaler, Daniel Yaqubi, <a href="https://arxiv.org/abs/1802.05990">Some determinants of path generating functions, II</a>, arXiv:1802.05990 [math.CO], 2018, Adv. Appl. Math. 101 (2018), 232-265.
%F a(2n) = C(2n+2, n), a(2n+1) = 2*a(2n).
%F E.g.f.: dif(Bessel_I(1,2x)+2*Bessel_I(2,2x)+Bessel_I(3,2x),x). - _Paul Barry_, Jun 09 2007
%F 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
%F 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
%t Module[{r=Range[0,20],b},Riffle[b=Binomial[2r+2,r],2b]] (* _Paolo Xausa_, Dec 14 2023 *)
%Y Cf. A001791, A162551 (bisections).
%K nonn
%O 0,2
%A _Clark Kimberling_
%E Definition corrected by _Herbert Kociemba_, May 08 2004
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