A024997 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0 = s(n), |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3. Also a(n) = T(n,n), where T is the array defined in A024996.

2, 8, 20, 58, 162, 462, 1318, 3782, 10886, 31436, 91016, 264134, 768094, 2237640, 6529284, 19079574, 55826166, 163538472, 479588844, 1407813438, 4136307798, 12163015662, 35793391662, 105407889930, 310620540202, 915913267652, 2702265079208

Offset: 3

Author

Clark Kimberling

Keywords

• nonn

Comments

Second differences of the central trinomial coefficients A002426. - T. D. Noe, Mar 16 2005

Links

• G. C. Greubel, Table of n, a(n) for n = 3..1000

Crossrefs

Cf. A025179.

Programs

Mathematica

Rest[Differences[CoefficientList[Series[1/Sqrt[(1 + x) (1 - 3 x)], {x, 0, 30}], x], 2]] (* Harvey P. Dale, May 11 2013 *)
Table[2 Sum[Binomial[n, 2 k] Binomial[2 k + 1, k + 1], {k, 0, Floor[n/2]}],  {n, 1, 25}] (* G. C. Greubel, Mar 01 2017 *)
Rest[Rest[CoefficientList[Series[((1 - x)^2 - (1 - x) Sqrt[1 - 2 x - 3 x^2])/(x Sqrt[1 - 2 x - 3 x^2]), {x, 0, 15}], x]]] (* G. C. Greubel, Mar 02 2017 *)

PARI

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

Formula

a(n) = 2*A025179(n-1).

From G. C. Greubel, Mar 01 2017: (Start)

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

O.g.f.: ((1-x)^2-(1-x+2*x^2)*sqrt(1-2*x-3*x^2)) / sqrt(1-2*x-3*x^2) [corrected by Charles R Greathouse IV, Mar 05 2017]

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