OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..3134 (terms 0..1000 from Alois P. Heinz)
FORMULA
G.f.: 1/Product_{m>0} (1 - m^2*x^m).
Recurrence: a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d^(2*k/d + 1).
a(n) = S(n,1), where S(n,m) = n^2 + Sum_{k=m..n/2} k^2*S(n-k,k), S(n,n) = n^2, S(n,m) = 0 for m > n. - Vladimir Kruchinin, Sep 07 2014
From Vaclav Kotesovec, Mar 16 2015: (Start)
a(n) ~ c * 3^(2*n/3), where
c = 668.1486183948153029651700839617715291485899132694809388646986235... if n=3k
c = 667.8494657534167286226227360927068283390090685342574808235616845... if n=3k+1
c = 667.8481656987523944806949678900876994934226621916594805916358627... if n=3k+2
(End)
In closed form, a(n) ~ (Product_{k>=4}(1/(1 - k^2/3^(2*k/3))) / ((1 - 3^(-2/3)) * (1 - 4*3^(-4/3))) + Product_{k>=4}(1/(1 - (-1)^(2*k/3)*k^2/3^(2*k/3))) / ((-1)^(2*n/3) * (1 + 4/3*(-1/3)^(1/3)) * (1 - (-1/3)^(2/3))) + Product_{k>=4}(1/(1 - (-1)^(4*k/3)*k^2/3^(2*k/3))) / ((-1)^(4*n/3) * (1 + (-1)^(1/3)*3^(-2/3)) * (1 - 4*(-1)^(2/3)*3^(-4/3)))) * 3^(2*n/3 - 1). - Vaclav Kotesovec, Apr 25 2017
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018
EXAMPLE
The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products of squares of parts are 16,9,16,4,1 and their sum is a(4) = 46.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Sep 07 2014
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, i^2*b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 02 2015, after Alois P. Heinz *)
Table[Total[Times@@(#^2)&/@IntegerPartitions[n]], {n, 0, 30}] (* Harvey P. Dale, Apr 29 2018 *)
Table[Total[Times@@@(IntegerPartitions[n]^2)], {n, 0, 30}] (* Harvey P. Dale, Sep 07 2023 *)
PROG
(Maxima)
S(n, m):=if n=0 then 1 else if n<m then 0 else if n=m then n^2 else sum(k^2*S(n-k, k), k, m, n/2)+n^2;
makelist(S(n, 1), n, 1, 27); /* Vladimir Kruchinin, Sep 07 2014 */
(PARI) N=22; q='q+O('q^N); Vec(1/prod(n=1, N, 1-n^2*q^n)) \\ Joerg Arndt, Aug 31 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Nov 30 2002
STATUS
approved