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%I #15 Jun 05 2021 06:28:34
%S 0,1,7,23,66,145,321,600,1137,1964,3379,5463,8888,13714,21206,31737,
%T 47319,68727,99718,141488,200383,279097,387302,530286,724113,976949,
%U 1314106,1751079,2325412,3062942,4022617,5244455,6817630,8808369,11346219,14536656,18573495
%N Sum of the second moments of all partitions of n.
%C The first element of each partition is given weight 1.
%H Alois P. Heinz, <a href="/A066187/b066187.txt">Table of n, a(n) for n = 0..1000</a>
%e a(3) = 23 because the second moments of all partitions of 3 are {3}.{1},{2,1}.{1,4} and {1,1,1}.{1,4,9}, resulting in 3,6,14; summing to 23.
%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, 1)[2]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 29 2014
%t Table[ Plus@@Map[ #.Range[ 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 # + {0, #[[1]]*i*t^2}& @ b[n - i, i, t + 1]]]];
%t a[n_] := b[n, n, 1][[2]];
%t a /@ Range[0, 50] (* _Jean-François Alcover_, Jun 05 2021, after _Alois P. Heinz_ *)
%K easy,nonn
%O 0,3
%A _Wouter Meeussen_, Dec 15 2001
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