A026165 Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 2, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T in A026148.

1, 2, 6, 17, 49, 141, 407, 1177, 3411, 9904, 28808, 83931, 244895, 715534, 2093262, 6130767, 17974779, 52751358, 154950378, 455524203, 1340182539, 3945723033, 11624603235, 34268836707, 101081770181, 298320243976, 880875609552

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k+1, floor(k/2)). - Vladeta Jovovic, Sep 18 2003

E.g.f.: exp(x)/x^2*((2*x^2-2*x)*BesselI(0, 2*x)+(2-x+2*x^2)*BesselI(1, 2*x)). - Vladeta Jovovic, Sep 23 2003

Conjecture: (n+3)*a(n) + (-5*n-9)*a(n-1) + (5*n+1)*a(n-2) + 5*(n-1)*a(n-3) + 6*(-n+3)*a(n-4) = 0. - R. J. Mathar, Jun 23 2013

Recurrence: (n+3)*(2*n^2 - n + 1)*a(n) = (4*n^3 + 10*n^2 + 7*n - 5)*a(n-1) + 3*(n-1)*(2*n^2 + 3*n + 2)*a(n-2). - Vaclav Kotesovec, Feb 01 2014

a(n) ~ 2 * 3^(n+1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 01 2014

Mathematica

CoefficientList[Series[E^x/x^2*((2*x^2-2*x)*BesselI[0, 2*x]+(2-x+2*x^2)*BesselI[1, 2*x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec after Vladeta Jovovic, Feb 01 2014 *)

Crossrefs

Sequence in context: A052536 A122100 A122099 * A336742 A148445 A148446

Adjacent sequences: A026162 A026163 A026164 * A026166 A026167 A026168

Keyword

nonn

Author

Clark Kimberling

Status

approved