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Sum of products of squares of parts in all partitions of n.
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%I #44 Sep 07 2023 14:30:02

%S 1,1,5,14,46,107,352,789,2314,5596,14734,34572,92715,210638,531342,

%T 1250635,3042596,6973974,16973478,38399806,91301956,207992892,

%U 483244305,1089029008,2533640066,5642905974,12912848789,28893132440,65342580250,144803524640

%N Sum of products of squares of parts in all partitions of n.

%H Seiichi Manyama, <a href="/A077335/b077335.txt">Table of n, a(n) for n = 0..3134</a> (terms 0..1000 from Alois P. Heinz)

%F G.f.: 1/Product_{m>0} (1 - m^2*x^m).

%F 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).

%F 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

%F From _Vaclav Kotesovec_, Mar 16 2015: (Start)

%F a(n) ~ c * 3^(2*n/3), where

%F c = 668.1486183948153029651700839617715291485899132694809388646986235... if n=3k

%F c = 667.8494657534167286226227360927068283390090685342574808235616845... if n=3k+1

%F c = 667.8481656987523944806949678900876994934226621916594805916358627... if n=3k+2

%F (End)

%F 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

%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 14 2018

%e 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.

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 07 2014

%t 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_ *)

%t Table[Total[Times@@(#^2)&/@IntegerPartitions[n]],{n,0,30}] (* _Harvey P. Dale_, Apr 29 2018 *)

%t Table[Total[Times@@@(IntegerPartitions[n]^2)],{n,0,30}] (* _Harvey P. Dale_, Sep 07 2023 *)

%o (Maxima)

%o 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;

%o makelist(S(n,1),n,1,27); /* _Vladimir Kruchinin_, Sep 07 2014 */

%o (PARI) N=22;q='q+O('q^N); Vec(1/prod(n=1,N,1-n^2*q^n)) \\ _Joerg Arndt_, Aug 31 2015

%Y Cf. A006906, A074141, A092484, A265844, A292164.

%K nonn

%O 0,3

%A _Vladeta Jovovic_, Nov 30 2002