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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)).

%I A273020 #19 Mar 01 2020 07:45:28
%S A273020 1,1,3,5,19,39,141,321,1107,2675,8953,22483,73789,190345,616227,
%T A273020 1621413,5196627,13882947,44152809,119385663,377379369,1030434069,
%U A273020 3241135527,8921880135,27948336381,77459553549,241813226151,674100041501,2098240353907,5878674505303,18252025766941
%N 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)).
%H A273020 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 A273020 a(n) = JacobiP(n, 1, -n-3/2, -7)/(n+1) + GegenbauerC(n-1, -n, -1/2), with a(0) = 1.
%F A273020 a(n) = hypergeom([-n,1/2], [2], 4) + n*hypergeom([-n/2+1,-n/2+1/2], [2], 4).
%F A273020 a(n) = (-1)^n*A005043(n) + A005717(n).
%F A273020 a(2*n) = A082758(n).
%F A273020 a(2*n+1) = A273019(n).
%p A273020 seq(simplify(hypergeom([-n,1/2],[2],4) + n*hypergeom([-n/2+1,-n/2+1/2],[2],4)), n=0..30);
%t A273020 Table[ JacobiP[n, 1, -n-3/2, -7]/(n+1) + GegenbauerC[n-1,-n,-1/2], {n,0,30} ]
%o A273020 (Sage)
%o A273020 def A():
%o A273020     a, b, c, d, n = 0, 1, 1, -1, 1
%o A273020     yield 1
%o A273020     while True:
%o A273020         yield d + b*(1-(-1)^n)
%o A273020         n += 1
%o A273020         a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
%o A273020         c, d = d, (3*(n-1)*c-(2*n-1)*d)//n
%o A273020 A273020 = A()
%o A273020 print([next(A273020) for _ in range(31)])
%Y A273020 Cf. A005043, A005717, A082758, A273019.
%K A273020 nonn
%O A273020 0,3
%A A273020 _Peter Luschny_, May 13 2016