A026016 a(n) = binomial(2*n-1, n) - binomial(2*n-1, n+3).

1, 3, 10, 34, 117, 407, 1430, 5070, 18122, 65246, 236436, 861764, 3157325, 11622015, 42961470, 159419670, 593636670, 2217608250, 8308432140, 31212003420, 117544456770, 443690433654, 1678353186780, 6361322162444, 24155384502452, 91882005146652

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Number of (s(0), s(1), ..., s(2n-1)) 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-1) = 3. Also a(n) = T(2n-1,n-1), where T is the array defined in A026009.

Number of integer lattice paths from (0,2) to (n-1,n+2) that do not cross the main diagonal.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000108, A026009.

Programs

Magma

[Binomial(2*n-1, n) - Binomial(2*n-1, n+3): n in [1..30]]; // Vincenzo Librandi, Jun 21 2013

Maple

a:= n-> binomial(2*n-1, n) -binomial(2*n-1, n+3): seq(a(n), n=1..27); #  Zerinvary Lajos, Dec 10 2007

Mathematica

Table[Binomial[2 n - 1, n] - Binomial[2 n - 1, n + 3], {n, 1, 40}] (* Vincenzo Librandi, Jun 21 2013 *)

PARI

a(n) = binomial(2*n-1, n) - binomial(2*n-1, n+3) \\ Charles R Greathouse IV, Oct 23 2023

Sage

[binomial(2*n-1, n) - binomial(2*n-1, n+3) for n in (2..30)] # G. C. Greubel, Mar 19 2021

Formula

Expansion of (1+x^1*C^3)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

(n+3)*a(n) +(-7*n-9)*a(n-1) +2*(7*n-4)*a(n-2) +4*(-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 -4*x +2*x^2 - (1-2*x)*sqrt(1-4*x) )/(2*x^3).

E.g.f.: -1 + (exp(2*x)/x^2)*( x*(1+x)*BesselI(0, 2*x) - (2 +x +2*x^2)*Bessel(1, 2*x) ).

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

a(n) = C(n+2) -3*C(n+1) +2*C(n) = 6*((n^2+1)/((n+2)*(n+3)))*C(n). - G. C. Greubel, Mar 22 2021

Extensions

Better description from Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001