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

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A024997, A026135, A359364.

Programs

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 *)

PARI

my(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

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