OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..79
FORMULA
a(n) = [x^n] Product_{k=1..n} 1/(1 + k^n*x^k).
MAPLE
b:= proc(n, i, k) option remember; (m->
`if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
`if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
end:
g:= proc(n, k) option remember; `if`(n=0, 1,
-add(b(n-i$2, k)*g(i, k), i=0..n-1))
end:
a:= n-> g(n$2):
seq(a(n), n=0..15); # Alois P. Heinz, Sep 12 2017
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = Function[m, If[m < n, 0, If[n == m, i!^k, b[n, i - 1, k] + If[i > n, 0, i^k*b[n - i, i - 1, k]]]]][i*(i + 1)/2];
g[n_, k_] := g[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k]*g[i, k], {i, 0, n-1}]];
a[n_] := g[n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 03 2018, after Alois P. Heinz *)
PROG
(PARI) {a(n) = polcoeff(1/prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}
(Python)
from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, k):
m=i*(i + 1)/2
return 0 if m<n else f(i)**k if n==m else b(n, i - 1, k) + (0 if i>n else i**k*b(n - i, i - 1, k))
@cacheit
def g(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*g(i, k) for i in range(n)])
def a(n): return g(n, n)
print([a(n) for n in range(16)]) # Indranil Ghosh, Sep 14 2017, after Maple program
KEYWORD
sign
AUTHOR
Seiichi Manyama, Sep 12 2017
STATUS
approved