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A271370 Total number of inversions in all partitions of n.

%I A271370 #16 Feb 03 2017 13:16:52
%S A271370 0,0,0,1,3,9,18,38,68,120,200,326,508,785,1179,1741,2532,3633,5141,
%T A271370 7199,9972,13680,18618,25116,33642,44738,59139,77653,101444,131751,
%U A271370 170320,219049,280553,357652,454254,574507,724135,909265,1138169,1419737,1765884,2189441
%N A271370 Total number of inversions in all partitions of n.
%H A271370 Alois P. Heinz, <a href="/A271370/b271370.txt">Table of n, a(n) for n = 0..1000</a>
%H A271370 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion (discrete mathematics)</a>
%H A271370 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A271370 a(n) = Sum_{k>0} k * A264033(n,k).
%e A271370 a(3) = 1: one inversion in 21.
%e A271370 a(4) = 3: one inversion in 31, and two inversions in 211.
%p A271370 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A271370       add((p-> p+[0, p[1]*j*t])(b(n-i*j, i-1, t+j)), j=0..n/i)))
%p A271370     end:
%p A271370 a:= n-> b(n$2, 0)[2]:
%p A271370 seq(a(n), n=0..60);
%t A271370 b[n_, i_, t_] := b[n, i, t] = If[n==0, {1, 0}, If[i<1, 0, Sum[Function[p, If[p === 0, 0, p+{0, p[[1]]*j*t}]][b[n-i*j, i-1, t+j]], {j, 0, n/i}]]];
%t A271370 a[n_] := b[n, n, 0][[2]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 03 2017, translated from Maple *)
%Y A271370 Cf. A000041, A189052, A264033, A264082, A271371.
%K A271370 nonn
%O A271370 0,5
%A A271370 _Alois P. Heinz_, Apr 05 2016