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A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).

%I A092484 #31 Sep 07 2023 15:51:25
%S A092484 1,1,4,13,25,77,161,393,726,2010,3850,7874,16791,31627,69695,139560,
%T A092484 255997,482277,986021,1716430,3544299,6507128,11887340,21137849,
%U A092484 38636535,70598032,123697772,233003286,412142276,711896765,1252360770
%N A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).
%C A092484 Sum of squares of products of terms in all partitions of n into distinct parts.
%H A092484 Alois P. Heinz, <a href="/A092484/b092484.txt">Table of n, a(n) for n = 0..10000</a>
%F A092484 G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(2*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 14 2018
%F A092484 Conjecture: log(a(n)) ~ sqrt(2*n) * (log(2*n) - 2). - _Vaclav Kotesovec_, Dec 27 2020
%e A092484 The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding squares of products are 36, 25, 64, 36 and their sum is a(6) = 161.
%p A092484 b:= proc(n, i) option remember; (m->
%p A092484       `if`(m<n, 0, `if`(n=m, i!^2, b(n, i-1)+
%p A092484       `if`(i>n, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
%p A092484     end:
%p A092484 a:= n-> b(n$2):
%p A092484 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 10 2017
%t A092484 Take[ CoefficientList[ Expand[ Product[1 + m^2*q^m, {m, 100}]], q], 31] (* _Robert G. Wilson v_, Apr 05 2005 *)
%o A092484 (PARI) N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+n^2*x^n)) \\ _Seiichi Manyama_, Sep 10 2017
%Y A092484 Cf. A022629, A077335, A265844, A285737, A292165.
%Y A092484 Column k=2 of A292189.
%K A092484 nonn
%O A092484 0,3
%A A092484 _Jon Perry_, Apr 04 2004
%E A092484 More terms from _Robert G. Wilson v_, Apr 05 2004