A026013 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) = 4. Also a(n) = T(2n,n-1), where T is the array defined in A026009.

1, 4, 15, 55, 200, 726, 2639, 9620, 35190, 129200, 476102, 1760673, 6533150, 24319050, 90795375, 339929880, 1275977670, 4801199400, 18106714050, 68430306750, 259129680264, 983085703116, 3736124441990, 14222020085880, 54221213973500

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

G.f.: (x + x^2*C^3)*C^3 where C = g.f. for Catalan numbers A000108.

E.g.f.: exp(2*x)*(Bessel_I(1,2*x)-Bessel_I(4,2*x)). - Paul Barry, Jun 04 2007

Conjecture: (n+4)*(n+1)*a(n) -4*(n^2+4*n+6)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 13 2012

a(n) ~ 15 * 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019

From G. C. Greubel, Mar 18 2021: (Start)

a(n) = C(n+3) - 4*C(n+2) + 3*C(n+1) + Sum_{j=0..n-1} C(j+1)*C(n-j-1), where C(n) are the Catalan numbers (A000108).

a(n) = binomial(2*n, n-1) - binomial(2*n, n-4) = A026009(2*n, n-1). (End)

Mathematica

Rest[CoefficientList[Series[(-1 + Sqrt[1 - 4*x])^3 * (1 - x) * (-1 + Sqrt[1 - 4*x] + 2*x) / (16*x^4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Sep 03 2019 *)
With[{f = CatalanNumber}, Table[f[n+3] -4*f[n+2] +3*f[n+1] +Sum[f[j+1]*f[n-j-1], {j, 0, n-1}], {n, 30}]] (* G. C. Greubel, Mar 18 2021 *)

Prog

(Sage) [binomial(2*n, n-1) - binomial(2*n, n-4) for n in (1..30)] # G. C. Greubel, Mar 18 2021
(Magma) [Binomial(2*n, n-1) - Binomial(2*n, n-4): n in [1..30]]; // G. C. Greubel, Mar 18 2021

Crossrefs

Cf. A000108, A026009.

Sequence in context: A094833 A039717 A220948 * A371820 A050183 A094375

Adjacent sequences: A026010 A026011 A026012 * A026014 A026015 A026016

Keyword

nonn

Author

Clark Kimberling

Extensions

Corrected by Franklin T. Adams-Watters, Oct 25 2006

Status

approved