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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 23 10:45 EST 2020. Contains 331171 sequences. (Running on oeis4.)