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

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

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

Sequence in context: A174935 A270620 A270636 * A176640 A126729 A336006

Adjacent sequences: A077332 A077333 A077334 * A077336 A077337 A077338

Keyword

nonn

Author

Vladeta Jovovic, Nov 30 2002

Status

approved