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%I #18 Feb 04 2024 22:28:04
%S 1,2,3,6,36,288,3360,55440,1241856,36427776,1358235648,62818398720,
%T 3531789972480,237336286150656,18792718657929216,1732062236305809408,
%U 183865068161693614080,22273939685873740677120
%N a(n) = sum of sums of all sets of three distinct preceding terms otherwise a(n) = n for n <= 3.
%H G. C. Greubel, <a href="/A129379/b129379.txt">Table of n, a(n) for n = 1..270</a>
%F a(n) = (1/2)*(n-2)*(n-3)*Sum_{j=1..n-1} a(j) for n > 3, with a(1) = 1, a(2) = 2, a(3) = 3.
%F a(n) = A000217(n-3)*A129380(n-1) for n > 3.
%F From _G. C. Greubel_, Feb 02 2024: (Start)
%F a(n) = (6/2^(n-4))*binomial(n-2,2)*|Pochhammer((3+i*sqrt(7))/2, n-4)|^2 for n > 3, otherwise a(n) = n.
%F a(n) = (3/2^(n-3))*binomial(n-2,2)*Product_{k=0..n-3} (k^2 - k + 2), for n > 3, otherwise a(n) = n.
%F a(n) = (n-2)*(n^2-7*n+14)/(2*(n-4))*a(n-1), for n > 4, otherwise a(n) = binomial(n, floor(n/2)).
%F (End)
%F From _Vaclav Kotesovec_, Feb 03 2024: (Start)
%F For n>=4, a(n) = 3 * 2^(3-n) * (n-3) * (n-2) * cosh(sqrt(7)*Pi/2) * Gamma(n - 5/2 - i*sqrt(7)/2) * Gamma(n - 5/2 + i*sqrt(7)/2)/Pi, where i is the imaginary unit.
%F a(n) ~ 3 * cosh(sqrt(7)*Pi/2) * n^(2*n-4) / (2^(n-4) * exp(2*n)). (End)
%t a[n_]:= a[n]= If[n<5, Binomial[n, Floor[n/2]], (n-2)*(n^2-7*n+14)*a[n- 1]/(2*(n-4))];
%t Table[a[n], {n,40}] (* _G. C. Greubel_, Feb 02 2024 *)
%t Round[Flatten[{{1, 2, 3}, Table[3 * 2^(3-n) * (n-3) * (n-2) * Cosh[Sqrt[7]*Pi/2] * Gamma[n - 5/2 - I*Sqrt[7]/2] * Gamma[n - 5/2 + I*Sqrt[7]/2]/Pi, {n, 4, 20}]}]] (* _Vaclav Kotesovec_, Feb 03 2024 *)
%o (Magma)
%o A129379:= func< n | n le 3 select Binomial(n,Floor(n/2)) else (3/2^(n-3))*Binomial(n-2,2)*(&*[k^2-k+2: k in [0..n-3]]) >;
%o [A129379(n): n in [1..30]]; // _G. C. Greubel_, Feb 02 2024
%o (SageMath)
%o def A129379(n): return binomial(n,n//2) if n<4 else 3*binomial(n-2,2)*product(j^2-j+2 for j in range(n-2))//2^(n-3)
%o [A129379(n) for n in range(1,31)] # _G. C. Greubel_, Feb 02 2024
%Y Cf. A000217, A083746, A129380.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Apr 14 2007
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