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%I #13 Mar 25 2024 11:18:56
%S 1,5,133,4037,129285,4266830,143567173,4896136845,168640510725,
%T 5853000551090,204368928058958,7170955214476509,252638095187722437,
%U 8931025389858103602,316640855103349347725,11254413331736554364987,400893874585938826203909,14307778459379093347171266
%N a(n) = binomial(4*n - 1, 2*n - 1)*hypergeom([-n, -n + 1/2], [2*n + 1], 4).
%C Related to the Motzkin triangle A064189 counting certain lattice paths.
%F a(n) = Sum_{j=0..4*n-1} C(4*n-1, j)*(C(4*n-1-j, j+2*n-1) - C(4*n-1-j, j+2*n+1)) for n >= 1.
%F a(n) = A064189(4*n - 1, 2*n - 1) for n >= 1.
%F a(n) = A344394(4*n - 1) for n >= 1.
%F a(n) ~ sqrt(1014 + 156*sqrt(13)) * (13688 + 3640*sqrt(13))^n / (52 * sqrt(Pi*n) * 3^(6*n+1)). - _Vaclav Kotesovec_, Feb 18 2024
%F D-finite with recurrence +9*n*(6*n-1)*(3*n-1)*(3835115277622*n -6057563812695) *(2*n-1)*(3*n-2) *(6*n-5)*a(n) +2*(776430552534185648*n^7 -13254965233720706112*n^6 +77698256107321929944*n^5 -233839293644869788720*n^4 +406279253239920624227*n^3 -412808144693534857728*n^2 +228023561050132883751*n -52874097275943488160)*a(n-1) -108*(4*n-5)*(4*n-7) *(51631651831183544*n^5 -528937515408392660*n^4 +2125620894576233062*n^3 -4194554621940993427*n^2 +4055650255694760927*n -1531029729082241880)*a(n-2) +402408*(4*n-11)*(n-2) *(4*n-5)*(4*n-9)*(330342177838*n -391995025711)*(2*n-5) *(4*n-7)*a(n-3)=0. - _R. J. Mathar_, Mar 25 2024
%p alias(C=binomial):
%p a := n -> `if`(n = 0, 1, add(C(4*n - 1, j)*(C(4*n - 1 - j, j + 2*n - 1) - C(4*n - 1 - j, j + 2*n + 1)), j = 0..4*n-1)): seq(a(n), n = 0..17);
%t a[n_] := Binomial[4 n - 1, 2 n - 1] Hypergeometric2F1[-n, -n + 1/2, 2 n + 1, 4];
%t Table[a[n], {n, 0, 19}]
%Y Cf. A064189, A344394, A327871.
%K nonn
%O 0,2
%A _Peter Luschny_, May 19 2021
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