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A292165 Expansion of Product_{k>=1} 1/(1 + k^2*x^k).

%I A292165 #21 Mar 21 2022 08:08:17
%S A292165 1,-1,-3,-6,6,5,40,11,226,-516,-186,-844,3731,-3734,814,-33819,85660,
%T A292165 -46022,210342,-411678,593996,-2980156,2076721,-3445584,40785410,
%U A292165 -37503158,98085,-271846888,336918770,-295108832,2178341296,-2404059340,6127604258
%N A292165 Expansion of Product_{k>=1} 1/(1 + k^2*x^k).
%H A292165 Alois P. Heinz, <a href="/A292165/b292165.txt">Table of n, a(n) for n = 0..3145</a>
%F A292165 Convolution inverse of A092484.
%F A292165 From _Vaclav Kotesovec_, Sep 10 2017: (Start)
%F A292165 a(n) ~ (-1)^n * c * 3^(2*n/3), where
%F A292165 c = 0.717271758899891528435966115495396784611147877234945... if mod(n,3)=0
%F A292165 c = 0.387695187106751505296020614217498222070185848125472... if mod(n,3)=1
%F A292165 c = 0.241939482775588594057384356004734639024152664456553... if mod(n,3)=2
%F A292165 (End)
%F A292165 G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*j^(2*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 18 2018
%p A292165 b:= proc(n, i) option remember; (m->
%p A292165       `if`(m<n, 0, `if`(n=m, i!^2, b(n, i-1)+
%p A292165       `if`(i>n, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
%p A292165     end:
%p A292165 a:= proc(n) option remember; `if`(n=0, 1,
%p A292165       -add(b(n-i$2)*a(i$2), i=0..n-1))
%p A292165     end:
%p A292165 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 10 2017
%t A292165 b[n_, i_] := b[n, i] = Function[m,
%t A292165      If[m < n, 0, If[n == m, i!^2, b[n, i - 1] +
%t A292165      If[i > n, 0, i^2*b[n - i, i - 1]]]]][i*(i + 1)/2];
%t A292165 a[n_] := a[n] = If[n == 0, 1, -Sum[b[n - i, n - i]*a[i], {i, 0, n - 1}]];
%t A292165 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 21 2022, after _Alois P. Heinz_ *)
%o A292165 (PARI) N=66; x='x+O('x^N); Vec(1/prod(n=1, N, 1+n^2*x^n))
%Y A292165 Cf. A077335, A092484.
%K A292165 sign
%O A292165 0,3
%A A292165 _Seiichi Manyama_, Sep 10 2017