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A292165 Expansion of Product_{k>=1} 1/(1 + k^2*x^k). 3


%S 1,-1,-3,-6,6,5,40,11,226,-516,-186,-844,3731,-3734,814,-33819,85660,

%T -46022,210342,-411678,593996,-2980156,2076721,-3445584,40785410,

%U -37503158,98085,-271846888,336918770,-295108832,2178341296,-2404059340,6127604258

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

%H Alois P. Heinz, <a href="/A292165/b292165.txt">Table of n, a(n) for n = 0..3145</a>

%F Convolution inverse of A092484.

%F From _Vaclav Kotesovec_, Sep 10 2017: (Start)

%F a(n) ~ (-1)^n * c * 3^(2*n/3), where

%F c = 0.717271758899891528435966115495396784611147877234945... if mod(n,3)=0

%F c = 0.387695187106751505296020614217498222070185848125472... if mod(n,3)=1

%F c = 0.241939482775588594057384356004734639024152664456553... if mod(n,3)=2

%F (End)

%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*j^(2*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 18 2018

%p b:= proc(n, i) option remember; (m->

%p `if`(m<n, 0, `if`(n=m, i!^2, b(n, i-1)+

%p `if`(i>n, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)

%p end:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p -add(b(n-i$2)*a(i$2), i=0..n-1))

%p end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 10 2017

%o (PARI) N=66; x='x+O('x^N); Vec(1/prod(n=1, N, 1+n^2*x^n))

%Y Cf. A077335, A092484.

%K sign

%O 0,3

%A _Seiichi Manyama_, Sep 10 2017

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Last modified December 14 03:31 EST 2019. Contains 329978 sequences. (Running on oeis4.)