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A066188 Sum of the second moments of all partitions of n. 1

%I #17 May 30 2021 11:20:03

%S 0,0,1,6,22,58,141,289,579,1054,1885,3161,5280,8371,13220,20183,30605,

%T 45178,66448,95546,136877,192759,270146,373387,514187,699429,947846,

%U 1272067,1700456,2254007,2977623,3904091,5102058,6625434,8575277,11037277,14163777

%N Sum of the second moments of all partitions of n.

%C The first element of each partition is given weight 0.

%H Vaclav Kotesovec, <a href="/A066188/b066188.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..1000 from Alois P. Heinz)

%F a(n) ~ exp(Pi*sqrt(2*n/3)) * n / (5*sqrt(3)). - _Vaclav Kotesovec_, May 30 2021

%e a(3) = 6 because the second moments of all partitions of 3 are {3}.{0},{2,1}.{0,1} and {1,1,1}.{0,1,4}, resulting in 0,1,5; summing to 6.

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

%p `if`(i<1, [0$2], `if`(i>n, b(n, i-1, t), b(n, i-1, t)+

%p (h-> h+[0, h[1]*i*t^2])(b(n-i, i, t+1)))))

%p end:

%p a:= n-> b(n$2, 0)[2]:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 29 2014

%t Table[ Plus@@Map[ #.Range[ 0, -1+Length[ # ] ]^2&, IntegerPartitions[ n ] ], {n, 30} ]

%t (* Second program: *)

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0},

%t If[i < 1, {0, 0}, If[i > n, b[n, i - 1, t], b[n, i - 1, t] +

%t Function[h, h + {0, h[[1]]*i*t^2}][b[n - i, i, t + 1]]]]];

%t a[n_] := b[n, n, 0][[2]];

%t a /@ Range[0, 50] (* _Jean-Fran├žois Alcover_, May 30 2021, after _Alois P. Heinz_ *)

%K easy,nonn

%O 0,4

%A _Wouter Meeussen_, Dec 15 2001

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Last modified July 13 14:24 EDT 2024. Contains 374284 sequences. (Running on oeis4.)