A092484 - OEIS

%I #31 Sep 07 2023 15:51:25

%S 1,1,4,13,25,77,161,393,726,2010,3850,7874,16791,31627,69695,139560,

%T 255997,482277,986021,1716430,3544299,6507128,11887340,21137849,

%U 38636535,70598032,123697772,233003286,412142276,711896765,1252360770

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

%C Sum of squares of products of terms in all partitions of n into distinct parts.

%H Alois P. Heinz, <a href="/A092484/b092484.txt">Table of n, a(n) for n = 0..10000</a>

%F 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 Conjecture: log(a(n)) ~ sqrt(2*n) * (log(2*n) - 2). - _Vaclav Kotesovec_, Dec 27 2020

%e 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 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:= n-> b(n$2):

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

%t Take[ CoefficientList[ Expand[ Product[1 + m^2*q^m, {m, 100}]], q], 31] (* _Robert G. Wilson v_, Apr 05 2005 *)

%o (PARI) N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+n^2*x^n)) \\ _Seiichi Manyama_, Sep 10 2017

%Y Cf. A022629, A077335, A265844, A285737, A292165.

%Y Column k=2 of A292189.

%K nonn

%O 0,3

%A _Jon Perry_, Apr 04 2004

%E More terms from _Robert G. Wilson v_, Apr 05 2004