A025179 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 1. Also a(n) = T(n,n-1), where T is the array defined in A025177.

1, 4, 10, 29, 81, 231, 659, 1891, 5443, 15718, 45508, 132067, 384047, 1118820, 3264642, 9539787, 27913083, 81769236, 239794422, 703906719, 2068153899, 6081507831, 17896695831, 52703944965, 155310270101, 457956633826, 1351132539604

Offset

2,2

Links

G. C. Greubel, Table of n, a(n) for n = 2..1000 (terms 2..200 from T. D. Noe)

Formula

Equals (1/2) * A024997(n+1).

From Vladeta Jovovic, Jan 01 2004: (Start)

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k+1, k+1).

E.g.f.: exp(x)*(BesselI(0, 2*x)+BesselI(2, 2*x)). (End)

From Paul Barry, Sep 17 2005: (Start)

G.f.: ((1-x)^2 - (1-x)*sqrt(1-2*x-3*x^2))/(2*x*sqrt(1-2*x-3*x^2)).

a(n+1) = Sum_{k=0..n} C(n, k)*C(k+1, k/2+1)*(1+(-1)^k)/2}. (End)

D-finite with recurrence (n+1)*a(n) +(-3*n+1)*a(n-1) +(-n-5)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 26 2012

a(n) ~ 3^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 13 2014

Prepend 1 to the data, assume offset 0, and denote the resulting sequence alpha. Then alpha(n) = Sum_{k=0..n} Sum_{j=0..k} A359364(n, n - j). - Peter Luschny, Jan 10 2023

Mathematica

Rest[Rest[CoefficientList[Series[((1-x)^2-(1-x)*Sqrt[1-2*x-3*x^2]) /(2*x*Sqrt[1-2*x-3*x^2]), {x, 0, 20}], x]]] (* Vaclav Kotesovec, Feb 13 2014 *)

Prog

(PARI) x='x +O('x^50); Vec(((1-x)^2-(1-x +2*x^2)*sqrt(1-2*x-3*x^2)) /(2*x*sqrt(1 - 2*x -3*x^2))) \\ G. C. Greubel, Mar 01 2017

Crossrefs

Cf. A024997, A026135, A359364.

Sequence in context: A329156 A052946 A026152 * A116388 A221420 A212262

Adjacent sequences: A025176 A025177 A025178 * A025180 A025181 A025182

Keyword

nonn

Author

Clark Kimberling

Status

approved