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A077335 Sum of products of squares of parts in all partitions of n.

%I A077335 #44 Sep 07 2023 14:30:02
%S A077335 1,1,5,14,46,107,352,789,2314,5596,14734,34572,92715,210638,531342,
%T A077335 1250635,3042596,6973974,16973478,38399806,91301956,207992892,
%U A077335 483244305,1089029008,2533640066,5642905974,12912848789,28893132440,65342580250,144803524640
%N A077335 Sum of products of squares of parts in all partitions of n.
%H A077335 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 A077335 G.f.: 1/Product_{m>0} (1 - m^2*x^m).
%F A077335 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 A077335 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 A077335 From _Vaclav Kotesovec_, Mar 16 2015: (Start)
%F A077335 a(n) ~ c * 3^(2*n/3), where
%F A077335 c = 668.1486183948153029651700839617715291485899132694809388646986235... if n=3k
%F A077335 c = 667.8494657534167286226227360927068283390090685342574808235616845... if n=3k+1
%F A077335 c = 667.8481656987523944806949678900876994934226621916594805916358627... if n=3k+2
%F A077335 (End)
%F A077335 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 A077335 G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 14 2018
%e A077335 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 A077335 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A077335       b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
%p A077335     end:
%p A077335 a:= n-> b(n$2):
%p A077335 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 07 2014
%t A077335 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 A077335 Table[Total[Times@@(#^2)&/@IntegerPartitions[n]],{n,0,30}] (* _Harvey P. Dale_, Apr 29 2018 *)
%t A077335 Table[Total[Times@@@(IntegerPartitions[n]^2)],{n,0,30}] (* _Harvey P. Dale_, Sep 07 2023 *)
%o A077335 (Maxima)
%o A077335 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 A077335 makelist(S(n,1),n,1,27); /* _Vladimir Kruchinin_, Sep 07 2014 */
%o A077335 (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 A077335 Cf. A006906, A074141, A092484, A265844, A292164.
%K A077335 nonn
%O A077335 0,3
%A A077335 _Vladeta Jovovic_, Nov 30 2002