OFFSET
1,7
COMMENTS
6-fold convolution of A000203.
Column k=6 of A319083.
In general, if the sequence "a" is a k-fold convolution of A000203, then Sum_{k=1..n} a(k) ~ Pi^(2*k) * n^(2*k) / (6^k * (2*k)!). - Vaclav Kotesovec, Sep 20 2024
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
FORMULA
Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / 22348298649600. - Vaclav Kotesovec, Sep 20 2024
MAPLE
b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
`if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 6):
seq(a(n), n=1..55); # Alois P. Heinz, Jul 26 2024
PROG
(Python)
from functools import lru_cache
from sympy import divisor_sigma
def A374979(n):
@lru_cache(maxsize=None)
def g(x):
f = factorint(x+1).items()
return(5*prod((p**(3*(e+1))-1)//(p**3-1) for p, e in f)-(5+6*x)*prod((p**(e+1)-1)//(p-1) for p, e in f))//12
return sum(g(i)*g(j)*g(n-3-i-j) for i in range(1, n-4) for j in range(1, n-i-3))
CROSSREFS
KEYWORD
nonn
AUTHOR
Chai Wah Wu, Jul 26 2024
STATUS
approved