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A292072 Main diagonal of A292068.

%I A292072 #31 May 15 2021 06:16:33
%S A292072 1,-1,-3,-20,66,4439,454420,4873175,-3803048954,-7320203267692,
%T A292072 -1403057989033446,6669491545211096686,78492109668913945526447,
%U A292072 69591502229308312804788424,-6243846072108996200105800383026,-604234376454072219680822138902122079
%N A292072 Main diagonal of A292068.
%H A292072 Alois P. Heinz, <a href="/A292072/b292072.txt">Table of n, a(n) for n = 0..79</a>
%F A292072 a(n) = [x^n] Product_{k=1..n} 1/(1 + k^n*x^k).
%p A292072 b:= proc(n, i, k) option remember; (m->
%p A292072       `if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
%p A292072       `if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
%p A292072     end:
%p A292072 g:= proc(n,k) option remember; `if`(n=0, 1,
%p A292072       -add(b(n-i$2, k)*g(i, k), i=0..n-1))
%p A292072     end:
%p A292072 a:= n-> g(n$2):
%p A292072 seq(a(n), n=0..15);  # _Alois P. Heinz_, Sep 12 2017
%t A292072 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];
%t A292072 g[n_, k_] := g[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k]*g[i, k], {i, 0, n-1}]];
%t A292072 a[n_] := g[n, n];
%t A292072 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jun 03 2018, after _Alois P. Heinz_ *)
%o A292072 (PARI) {a(n) = polcoeff(1/prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}
%o A292072 (Python)
%o A292072 from sympy.core.cache import cacheit
%o A292072 from sympy import factorial as f
%o A292072 @cacheit
%o A292072 def b(n, i, k):
%o A292072     m=i*(i + 1)/2
%o A292072     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))
%o A292072 @cacheit
%o A292072 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)])
%o A292072 def a(n): return g(n, n)
%o A292072 print([a(n) for n in range(16)]) # _Indranil Ghosh_, Sep 14 2017, after Maple program
%Y A292072 Cf. A292190, A292194.
%K A292072 sign
%O A292072 0,3
%A A292072 _Seiichi Manyama_, Sep 12 2017