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A292164 Expansion of Product_{k>=1} (1 - k^2*x^k).

%I A292164 #26 Feb 04 2024 12:40:45
%S A292164 1,-1,-4,-5,-7,27,17,167,110,-42,10,-706,-4001,-3915,3079,-18640,9869,
%T A292164 21403,130565,107250,-15661,420664,599540,-161785,-1232833,-5836888,
%U A292164 -5129796,6516714,-29068180,-14953045,-41490510,20261320,30395771,441235155,205289550
%N A292164 Expansion of Product_{k>=1} (1 - k^2*x^k).
%H A292164 Seiichi Manyama, <a href="/A292164/b292164.txt">Table of n, a(n) for n = 0..10000</a>
%F A292164 Convolution inverse of A077335.
%F A292164 G.f.: exp(-Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 18 2018
%p A292164 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A292164       b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
%p A292164     end:
%p A292164 a:= proc(n) option remember; `if`(n=0, 1,
%p A292164       -add(b(n-i$2)*a(i$2), i=0..n-1))
%p A292164     end:
%p A292164 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 10 2017
%t A292164 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A292164     b[n, i - 1] + If[i > n, 0, i^2*b[n - i, i]]]];
%t A292164 a[n_] := a[n] = If[n == 0, 1,
%t A292164     -Sum[b[n - i, n - i]*a[i], {i, 0, n - 1}]];
%t A292164 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 04 2024, after _Alois P. Heinz_ *)
%o A292164 (PARI) N=66; x='x+O('x^N); Vec(prod(n=1, N, 1-n^2*x^n))
%Y A292164 Column k=2 of A292166.
%Y A292164 Cf. A022629, A022661, A077335, A092484.
%K A292164 sign
%O A292164 0,3
%A A292164 _Seiichi Manyama_, Sep 10 2017