A292072 Main diagonal of A292068.

1, -1, -3, -20, 66, 4439, 454420, 4873175, -3803048954, -7320203267692, -1403057989033446, 6669491545211096686, 78492109668913945526447, 69591502229308312804788424, -6243846072108996200105800383026, -604234376454072219680822138902122079

Offset: 0

Author

Seiichi Manyama, Sep 12 2017

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..79

Crossrefs

Cf. A292190, A292194.

Programs

Maple

b:= proc(n, i, k) option remember; (m->
      `if`(mn, 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 *)

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 mn 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

Formula

a(n) = [x^n] Product_{k=1..n} 1/(1 + k^n*x^k).