A026014 a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 6. Also a(n) = T(2n,n-2), where T is the array defined in A026009.

1, 6, 28, 119, 483, 1911, 7448, 28764, 110466, 422807, 1615152, 6163885, 23514855, 89714835, 342411120, 1307613480, 4997082510, 19111589280, 73154916744, 280265589198, 1074685552094, 4124573481446, 15843809385168, 60914041121640

Offset

2,2

Links

G. C. Greubel, Table of n, a(n) for n = 2..1000

Formula

-(n-2)*(n+5)*(n+23)*a(n) +(-n^3+127*n^2+188*n-432)*a(n-1) +2*(n-1)*(2*n-3)*(5*n-24)*a(n-2) = 0. - R. J. Mathar, Jun 20 2013

From G. C. Greubel, Mar 19 2021: (Start)

G.f.: (1-x)*(1 -7*x +14*x^2 -7*x^3 -(1 -5*x +6*x^2 -x^3)*sqrt(1-4*x))/(2*x^5).

G.f.: (1-x)*x^2*C(x)^7, where C(x) is the g.f. of the Catalan numbers (A000108).

E.g.f.: exp(2*x)*(BesselI(2, 2*x) - BesselI(5, 2*x)).

a(n) = binomial(2*n, n-2) - binomial(2*n, n-5) = A026009(2*n, n-2).

a(n) = 1 if n = 2 else f(n) - f(n-1), where f(n) = Sum_{j=0..n-2} C(n-j-2)*(C(j+5) -4*C(j+4) +3*C(j+3)) and C(n) are the Catalan numbers. (End)

From G. C. Greubel, Mar 22 2021: (Start)

a(n) = C(n+4) -6*C(n+3) +11*C(n+2) -7*C(n+1) +C(n).

a(n) = 21*(n*(n-1)*(n^2+n+4)/((n+2)*(n+3)*(n+4)*(n+5)))*C(n), where C(n) are the Catalan numbers. (End)

Mathematica

Table[Binomial[2*n, n-2] - Binomial[2*n, n-5], {n, 2, 30}] (* G. C. Greubel, Mar 19 2021 *)

Prog

(Sage) [binomial(2*n, n-2) - binomial(2*n, n-5) for n in (2..30)] # G. C. Greubel, Mar 19 2021
(Magma) [Binomial(2*n, n-2) - Binomial(2*n, n-5): n in [2..30]]; // G. C. Greubel, Mar 19 2021

Crossrefs

First differences of A000588.

Cf. A000108, A026009.

Sequence in context: A089096 A229587 A230492 * A332208 A226853 A259307

Adjacent sequences: A026011 A026012 A026013 * A026015 A026016 A026017

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved