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%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-François Alcover_, May 30 2021, after _Alois P. Heinz_ *)
%K easy,nonn
%O 0,4
%A _Wouter Meeussen_, Dec 15 2001
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