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

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

Author

Ilya Gutkovskiy, Jul 01 2019

Keywords

• nonn

Links

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

Crossrefs

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

Cf. A000010 (comments on product formulas).

Programs

Mathematica

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

PARI

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

Python

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

Formula

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