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A077335
Sum of products of squares of parts in all partitions of n.
20
1, 1, 5, 14, 46, 107, 352, 789, 2314, 5596, 14734, 34572, 92715, 210638, 531342, 1250635, 3042596, 6973974, 16973478, 38399806, 91301956, 207992892, 483244305, 1089029008, 2533640066, 5642905974, 12912848789, 28893132440, 65342580250, 144803524640
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