A026015 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) = 8. Also a(n) = T(2n,n-3), where T is the array defined in A026009.

1, 8, 45, 219, 987, 4248, 17748, 72675, 293436, 1172908, 4653935, 18366075, 72186075, 282861360, 1105877880, 4316224860, 16825024134, 65525448960, 255024693434, 992116674142, 3858537980286, 15004402265424, 58343871881400

Offset

3,2

Links

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

Formula

-(n+6)*(n-3)*a(n) +2*(3*n^2+3*n-20)*a(n-1) +(-9*n^2+15*n+20)*a(n-2) +2*(n-2)*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Jun 20 2013

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

G.f.: (1-x)*((1-3*x)*(1 -6*x +9*x^2 -3*x^3) -(1-x)*(1 -6*x +9*x^2 -x^3)*sqrt(1-4*x))/(2*x^6).

G.f.: (1-x)*x^3*C(x)^9, where C(x) is the g.f. of the Catalan numbers (A000108).

E.g.f.: exp(2*x)*(BesselI(3, 2*x) - BesselI(6, 2*x)).

a(n) = binomial(2*n, n-3) - binomial(2*n, n-6) = A026009(2*n, n-3).

a(n) = f(n) - f(n-1), where f(n) = Sum_{j=0..n-3} C(n-j-3)*(C(j+7) -6*C(j+6) +10*C(j+5) -4*C(j+4)) and C(n) are the Catalan numbers. (End)

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

a(n) = C(n+5) -8*C(n+4) +22*C(n+3) -25*C(n+2) +11*C(n+1) -C(n).

a(n) = (9/20)*(binomial(n,3)/binomial(n+6,5))*(3*n^2 +3*n +20)*C(n). (End)

Mathematica

Table[Binomial[2*n, n-3] - Binomial[2*n, n-6], {n, 3, 30}] (* G. C. Greubel, Mar 19 2021 *)

Prog

(Sage) [binomial(2*n, n-3) - binomial(2*n, n-6) for n in (3..30)] # G. C. Greubel, Mar 19 2021
(Magma) [Binomial(2*n, n-3) - Binomial(2*n, n-6): n in [3..30]]; // G. C. Greubel, Mar 19 2021

Crossrefs

First differences of A001392.

Cf. A000108, A026009.

Sequence in context: A273267 A055222 A273305 * A002696 A016208 A216540

Adjacent sequences: A026012 A026013 A026014 * A026016 A026017 A026018

Keyword

nonn

Author

Clark Kimberling

Status

approved