A254865 a(n) = Product_{k = 1+n-floor(n/3) .. n} k.

1, 1, 3, 4, 5, 30, 42, 56, 504, 720, 990, 11880, 17160, 24024, 360360, 524160, 742560, 13366080, 19535040, 27907200, 586051200, 859541760, 1235591280, 29654190720, 43609104000, 62990928000, 1700755056000, 2506375872000, 3634245014400, 109027350432000, 160945136352000, 234102016512000, 7725366544896000, 11420107066368000

Offset: 1

Author

Antti Karttunen, Feb 09 2015

Keywords

• nonn

Comments

Informally: Take the upper third of natural numbers in range [1..n] and multiply them together.

Links

• Robert Israel, Table of n, a(n) for n = 1..1025

Crossrefs

Leftmost column of A254864.

Trisection: A064352.

Programs

Maple

seq(n!/(n-floor(n/3))!,n=1..50); # Robert Israel, Jul 15 2020

Mathematica

Array[#!/(# - Floor[#/3])! &, 34] (* Michael De Vlieger, Jul 15 2020 *)

PARI

a(n) = prod(k=1+n-n\3, n, k); \\ Michel Marcus, Jul 15 2020

Scheme

(define (A254865 n) (mul A000027 (+ 1 (- n (floor->exact (/ n 3)))) n))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
(define (A254865 n) (A254864bi n 1)) ;; Alternatively, using code given in A254864.

Formula

a(n) = Product_{k = 1+n-floor(n/3) .. n} k.

Other identities. For all n >= 1:

a(3n) = A064352(n).

From Robert Israel, Jul 15 2020: (Start) a(n) = n!/(n-floor(n/3))!.

a(3*k) = 3*k*a(3*k-1).

a(3*k+1) = (3*k+1)*a(3*k)/(2*k+1).

a(3*k+2) = (3*k+2)*a(3*k+1)/(2*k+2).

E.g.f.: (cosh(x^(3/2))-1)*(1+1/x) + sinh(x^(3/2))/sqrt(x).

(End)