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

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A000108, A026009.

Programs

Magma

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

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 *)

Sage

[binomial(2*n, n-1) - binomial(2*n, n-4) for n in (1..30)] # G. C. Greubel, Mar 18 2021

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)

Extensions

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