A371798 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-2*k-1,n-2*k).

1, 1, 2, 7, 26, 96, 356, 1331, 5014, 19006, 72412, 277058, 1063856, 4097510, 15823432, 61245987, 237536326, 922906150, 3591500972, 13996328322, 54614894396, 213360770840, 834409399672, 3266370155262, 12797894251276, 50184309630196, 196936674150296

Offset: 0

Author

Seiichi Manyama, Apr 06 2024

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Sergi Elizalde, Nadia Lafrenière, Joel Brewster Lewis, Erin McNicholas, Jessica Striker, and Amanda Welch, Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks, arXiv:2412.16368 [math.CO], 2024. See p. 13.

Crossrefs

Cf. A026641, A120305.

Programs

Mathematica

Table[Sum[(-1)^k Binomial[2n-2k-1,n-2k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Oct 31 2024 *)

PARI

a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-2*k-1, n-2*k));

Formula

a(n) = [x^n] 1/((1+x^2) * (1-x)^n).

a(n) = binomial(2*n-1, n)*hypergeom([1, (1-n)/2, -n/2], [1/2-n, 1-n], -1). - Stefano Spezia, Apr 06 2024

a(n) ~ 2^(2*n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 07 2024

Conjectured g.f.: 1 + x*(4 - 10*x + 8*x^2)/(2 - 11*x + 14*x^2 - 8*x^3 + (2 - 3*x)*sqrt(1 - 4*x)) (see Elizalde et al. at p. 13). - Stefano Spezia, Dec 27 2024