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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 28 18:29 EDT 2024. Contains 372919 sequences. (Running on oeis4.)