A098710 -id:A098710 - cp's OEIS frontend

A163087 Product{k|n} k$. Here '$' denotes the swinging factorial function (A056040).

1, 1, 2, 6, 12, 30, 240, 140, 840, 3780, 15120, 2772, 221760, 12012, 960960, 9266400, 10810800, 218790, 7351344000, 923780, 16761064320, 3259095840, 3910915008, 16224936, 41977154419200, 2028117000, 249864014400

Offset: 0

Peter Luschny, Jul 21 2009

nonn

			The set of positive divisors of 3 is {1,3}. Thus a(3) = 1$ * 3$ = 1 * 6 = 6.

Peter Luschny, Swinging Primes.

Cf. A098710, A163088, A163089.

a := proc(n) local i; mul(i,i=map(swing,numtheory[divisors](n))) end:

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Product[sf[k], {k, Divisors[n]}]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 26 2013 *)

A308943 a(n) = Product_{d|n} binomial(n,d).

Original entry on oeis.org

1, 2, 3, 24, 5, 1800, 7, 15680, 756, 113400, 11, 79693891200, 13, 4372368, 20495475, 44972928000, 17, 2028339316523520, 19, 52737518268864000, 3247700400, 3585005424, 23, 38135556819759802035135799296, 1328250, 87885070000, 370142004375, 10293527616645873600000, 29

Offset: 1

Ilya Gutkovskiy, Jul 01 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..419

Cf. A001142, A008578 (fixed points), A056045 (similar, with Sum), A098710, A135396.

Cf. A000010 (comments on product formulas).

Table[Product[Binomial[n, d], {d, Divisors[n]}], {n, 1, 29}]

a(n) = my(p=1); fordiv(n, d, p *= binomial(n, d)); p; \\ Michel Marcus, Jul 02 2019

from math import prod, comb
from sympy import divisors
def A308943(n): return prod(comb(n,d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 22 2024

a(n) = Product_{k=1..n} binomial(n,gcd(n,k))^(1/phi(n/gcd(n,k))) = Product_{k=1..n} binomial(n,n/gcd(n,k))^(1/phi(n/gcd(n,k))) where phi = A000010. - Richard L. Ollerton, Nov 08 2021

cp's OEIS Frontend

A163087 Product{k|n} k$. Here '$' denotes the swinging factorial function (A056040).

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