A254864 -id:A254864 - cp's OEIS frontend

A088487 a(n) = Sum_{k=1..8} floor(A254864(n,k)/A254864(n-1,k)), where A254864(n,k) = n! / (n-floor(n/3^k))!.

8, 10, 8, 8, 13, 8, 8, 24, 8, 8, 19, 8, 8, 22, 8, 8, 42, 8, 8, 28, 8, 8, 31, 8, 8, 86, 8, 8, 37, 8, 8, 40, 8, 8, 78, 8, 8, 46, 8, 8, 49, 8, 8, 96, 8, 8, 55, 8, 8, 58, 8, 8, 167, 8, 8, 64, 8, 8, 67, 8, 8, 132, 8, 8, 73, 8, 8, 76, 8, 8, 150, 8, 8, 82, 8, 8, 85, 8, 8, 328, 8, 8, 91, 8, 8, 94, 8, 8

Offset: 2

Roger L. Bagula, Nov 09 2003

Antti Karttunen, Table of n, a(n) for n = 2..10000

Cf. A254864, A088488.

p[n_, k_]=n!/Product[i, {i, 1, n-Floor[n/3^k]}] digits=200 f[n_]=Sum[Floor[p[n, k]/p[n-1, k]], {k, 1, 8}] at=Table[f[n], {n, 2, digits}]

A254864bi(n,k) = prod(i=(1+(n-(n\(3^k)))),n,i);
A088487(n) = sum(k=1,8,(A254864bi(n,k)\A254864bi(n-1,k)));
for(n=2, 10000, write("b088487.txt", n, " ", A088487(n)));

(define (A088487 n) (add (lambda (k) (floor->exact (/ (A254864bi n k) (A254864bi (- n 1) k)))) 1 8)) ;; Code for A254864bi given in A254864.
;; The following function implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=1..8} floor(A254864(n,k)/A254864(n-1,k)), where A254864(n,k) = n! / (n-floor(n/3^k))!.

Edited by Antti Karttunen, Feb 09 2015

A254876 Triangular table T(n,k) = n! / Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m, read by rows T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), ...

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 4, 6, 6, 6, 5, 6, 24, 24, 24, 30, 24, 120, 120, 120, 120, 84, 120, 720, 720, 720, 720, 720, 112, 720, 5040, 5040, 5040, 5040, 5040, 5040, 1008, 6480, 40320, 40320, 40320, 40320, 40320, 40320, 40320, 4320, 50400, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880

Offset: 1

Antti Karttunen, Feb 09 2015

An auxiliary array for computing A088488.

			The first rows of the triangular table:
1
1, 1
3, 2, 2
4, 6, 6, 6
5, 6, 24, 24, 24
30, 24, 120, 120, 120, 120
84, 120, 720, 720, 720, 720, 720
112, 720, 5040, 5040, 5040, 5040, 5040, 5040
1008, 6480, 40320, 40320, 40320, 40320, 40320, 40320, 40320
4320, 50400, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880
...

Antti Karttunen, Table of n, a(n) for n = 1..1275; the first 50 rows of triangular table

Cf. A002024, A002260, A088488, A254864.

A254876bi(n, k) = n! / prod(i=(n-((2*n)\(3^k))), (n-(n\(3^k))), i);

(define (A254876 n) (A254876bi (A002024 n) (A002260 n)))
(define (A254876bi n k) (/ (A000142 n) (mul A000027 (- n (floor->exact (/ (* 2 n) (expt 3 k)))) (- n (floor->exact (/ n (expt 3 k)))))))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))

T(n,k) = n! / Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m.

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

Original entry on oeis.org

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

Antti Karttunen, Feb 09 2015

nonn

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

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

Leftmost column of A254864.

Trisection: A064352.

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

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

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

(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.

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)

cp's OEIS Frontend

A088487 a(n) = Sum_{k=1..8} floor(A254864(n,k)/A254864(n-1,k)), where A254864(n,k) = n! / (n-floor(n/3^k))!.

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A254876 Triangular table T(n,k) = n! / Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m, read by rows T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), ...

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

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