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

3,1

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

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)

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 *)

Prog

(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

Crossrefs

Cf. A025179.

Sequence in context: A221066 A238760 A174477 * A081157 A099177 A100097

Adjacent sequences: A024994 A024995 A024996 * A024998 A024999 A025000

Keyword

nonn

Author

Clark Kimberling

Status

approved