A273020 - OEIS

%I #19 Mar 01 2020 07:45:28

%S 1,1,3,5,19,39,141,321,1107,2675,8953,22483,73789,190345,616227,

%T 1621413,5196627,13882947,44152809,119385663,377379369,1030434069,

%U 3241135527,8921880135,27948336381,77459553549,241813226151,674100041501,2098240353907,5878674505303,18252025766941

%N 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)).

%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 (2008).

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

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

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

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

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

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

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

%o (Sage)

%o def A():

%o     a, b, c, d, n = 0, 1, 1, -1, 1

%o     yield 1

%o     while True:

%o         yield d + b*(1-(-1)^n)

%o         n += 1

%o         a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))

%o         c, d = d, (3*(n-1)*c-(2*n-1)*d)//n

%o A273020 = A()

%o print([next(A273020) for _ in range(31)])

%Y Cf. A005043, A005717, A082758, A273019.

%K nonn

%O 0,3

%A _Peter Luschny_, May 13 2016