A273020 a(n) = Sum_{k=0..n} C(n,k)*((-1)^n*(C(k,n-k)-C(k,n-k-1))+C(n-k,k+1)).

1, 1, 3, 5, 19, 39, 141, 321, 1107, 2675, 8953, 22483, 73789, 190345, 616227, 1621413, 5196627, 13882947, 44152809, 119385663, 377379369, 1030434069, 3241135527, 8921880135, 27948336381, 77459553549, 241813226151, 674100041501, 2098240353907, 5878674505303, 18252025766941

Offset: 0

Author

Peter Luschny, May 13 2016

Keywords

• nonn

Links

• A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 (2008).

Crossrefs

Cf. A005043, A005717, A082758, A273019.

Programs

Maple

seq(simplify(hypergeom([-n,1/2],[2],4) + n*hypergeom([-n/2+1,-n/2+1/2],[2],4)), n=0..30);

Mathematica

Table[ JacobiP[n, 1, -n-3/2, -7]/(n+1) + GegenbauerC[n-1,-n,-1/2], {n,0,30} ]

Sage

def A():
    a, b, c, d, n = 0, 1, 1, -1, 1
    yield 1
    while True:
        yield d + b*(1-(-1)^n)
        n += 1
        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
        c, d = d, (3*(n-1)*c-(2*n-1)*d)//n
A273020 = A()
print([next(A273020) for _ in range(31)])

Formula

a(n) = JacobiP(n, 1, -n-3/2, -7)/(n+1) + GegenbauerC(n-1, -n, -1/2), with a(0) = 1.

a(n) = hypergeom([-n,1/2], [2], 4) + n*hypergeom([-n/2+1,-n/2+1/2], [2], 4).

a(n) = (-1)^n*A005043(n) + A005717(n).

a(2*n) = A082758(n).

a(2*n+1) = A273019(n).