|
|
A092484
|
|
Expansion of Product_{m>=1} (1 + m^2*q^m).
|
|
12
|
|
|
1, 1, 4, 13, 25, 77, 161, 393, 726, 2010, 3850, 7874, 16791, 31627, 69695, 139560, 255997, 482277, 986021, 1716430, 3544299, 6507128, 11887340, 21137849, 38636535, 70598032, 123697772, 233003286, 412142276, 711896765, 1252360770
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Sum of squares of products of terms in all partitions of n into distinct parts.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018
Conjecture: log(a(n)) ~ sqrt(2*n) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020
|
|
EXAMPLE
|
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.
|
|
MAPLE
|
b:= proc(n, i) option remember; (m->
`if`(m<n, 0, `if`(n=m, i!^2, b(n, i-1)+
`if`(i>n, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
end:
a:= n-> b(n$2):
|
|
MATHEMATICA
|
Take[ CoefficientList[ Expand[ Product[1 + m^2*q^m, {m, 100}]], q], 31] (* Robert G. Wilson v, Apr 05 2005 *)
|
|
PROG
|
(PARI) N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+n^2*x^n)) \\ Seiichi Manyama, Sep 10 2017
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|