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%I #15 Jul 16 2020 02:40:19
%S 1,1,3,4,5,30,42,56,504,720,990,11880,17160,24024,360360,524160,
%T 742560,13366080,19535040,27907200,586051200,859541760,1235591280,
%U 29654190720,43609104000,62990928000,1700755056000,2506375872000,3634245014400,109027350432000,160945136352000,234102016512000,7725366544896000,11420107066368000
%N a(n) = Product_{k = 1+n-floor(n/3) .. n} k.
%C Informally: Take the upper third of natural numbers in range [1..n] and multiply them together.
%H Robert Israel, <a href="/A254865/b254865.txt">Table of n, a(n) for n = 1..1025</a>
%F a(n) = Product_{k = 1+n-floor(n/3) .. n} k.
%F Other identities. For all n >= 1:
%F a(3n) = A064352(n).
%F From _Robert Israel_, Jul 15 2020: (Start) a(n) = n!/(n-floor(n/3))!.
%F a(3*k) = 3*k*a(3*k-1).
%F a(3*k+1) = (3*k+1)*a(3*k)/(2*k+1).
%F a(3*k+2) = (3*k+2)*a(3*k+1)/(2*k+2).
%F E.g.f.: (cosh(x^(3/2))-1)*(1+1/x) + sinh(x^(3/2))/sqrt(x).
%F (End)
%p seq(n!/(n-floor(n/3))!,n=1..50); # _Robert Israel_, Jul 15 2020
%t Array[#!/(# - Floor[#/3])! &, 34] (* _Michael De Vlieger_, Jul 15 2020 *)
%o (Scheme)
%o (define (A254865 n) (mul A000027 (+ 1 (- n (floor->exact (/ n 3)))) n))
%o (define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
%o (define (A254865 n) (A254864bi n 1)) ;; Alternatively, using code given in A254864.
%o (PARI) a(n) = prod(k=1+n-n\3, n, k); \\ _Michel Marcus_, Jul 15 2020
%Y Leftmost column of A254864.
%Y Trisection: A064352.
%K nonn
%O 1,3
%A _Antti Karttunen_, Feb 09 2015
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