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%I #11 May 19 2021 17:32:46
%S 1,1,2,5,9,25,44,133,230,726,1242,4037,6853,22737,38376,129285,217242,
%T 740554,1239980,4266830,7123765,24701425,41141916,143567173,238637282,
%U 837212650,1389206210,4896136845,8112107475,28703894775,47495492400,168640510725,278722764954
%N a(n) = binomial(n, n/2 - 1/4 + (-1)^n/4)*hypergeom([-n/4 - 1/8 + (-1)^n/8, -n/4 + 3/8 + (-1)^n/8], [n/2 + 7/4 + (-1)^n/4], 4).
%C Related to the Motzkin triangle A064189 counting certain lattice paths.
%F a(n) = Sum_{j = 0..n} C(n, j)*(C(n - j, j + n/2 - 1/4 + (-1)^n/4) - C(n - j, j + n/2 + 7/4 + (-1)^n/4).
%F a(n) = A064189(n, floor(n/2)), the middle column of the Motzkin triangle.
%F a(n) = A026300(n, ceiling(n/2)).
%p alias(C=binomial):
%p a := n -> add(C(n, j)*(C(n - j, j + n/2 - 1/4 + (-1)^n/4) - C(n - j, j + n/2 + 7/4 + (-1)^n/4)), j = 0..n): seq(a(n), n = 0..32);
%t a[n_] := Binomial[n, n/2 - 1/4 + (-1)^n/4] Hypergeometric2F1[-n/4 - 1/8 + (-1)^n/8, -n/4 + 3/8 + (-1)^n/8, n/2 + 7/4 + (-1)^n/4, 4];
%t Table[a[n], {n, 0, 32}]
%Y Cf. A026300, A064189, A026302 (even bisection), A344396 (odd bisection), A327871.
%K nonn
%O 0,3
%A _Peter Luschny_, May 19 2021
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