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%I #20 Aug 04 2023 10:14:32
%S 1,5,25,133,726,4037,22737,129285,740554,4266830,24701425,143567173,
%T 837212650,4896136845,28703894775,168640510725,992671051482,
%U 5853000551090,34562387229046,204368928058958,1209916827501876,7170955214476509,42543879586512435,252638095187722437
%N a(n) = binomial(2*n + 1, n)*hypergeom([-(n + 1)/2, -n/2], [n + 2], 4).
%C Related to the Motzkin triangle A064189 counting certain lattice paths.
%F a(n) = Sum_{j=0..2*n+1)} C(2*n + 1, j)*(C(2*n + 1 - j, j + n) - C(2*n + 1 - j, j + n + 2)).
%F a(n) = A064189(2*n+1, n).
%F a(n) = A026300(2*n+1, n+1).
%F a(n) ~ sqrt((5242 + 18674/sqrt(13))/2187) * ((70 + 26*sqrt(13))/27)^n / sqrt(Pi*n). - _Vaclav Kotesovec_, May 19 2021
%F From _Peter Bala_, Aug 03 2023: (Start)
%F P-recursive: 3*(13*n - 4)*(3*n + 2)*(3*n + 1)*(n + 1)*a(n) = 2*(2*n + 1)*(455*n^3 + 315*n^2 - 44*n - 24)*a(n-1) + 36*(13*n + 9)*(2*n + 1)*(2*n - 1)*n*a(n-2) with a(0) = 1 and a(1) = 5.
%F a(n) = (1/2)*A027908(n+1). (End)
%p alias(C=binomial):
%p a := n -> add(C(2*n + 1, j)*(C(2*n + 1 - j, j + n) - C(2*n + 1 - j, j + n + 2)), j = 0..2*n+1): seq(a(n), n=0..23);
%t a[n_] := Binomial[2 n + 1, n] Hypergeometric2F1[-(n + 1)/2, -n/2, n + 2, 4];
%t Table[a[n], {n, 0, 23}]
%Y Cf. A027908, A064189, A026300, A344394, A327871.
%K nonn
%O 0,2
%A _Peter Luschny_, May 19 2021
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