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A381373 Sum over all partitions of [n] of n^j for a partition with j inversions.

%I A381373 #22 Mar 15 2025 18:44:34
%S A381373 1,1,2,7,72,3276,915848,2011878835,42723411900032,
%T A381373 10608257527069388539,35808039364308986083608352,
%U A381373 1828963737334508176477805993389490,1618534282345584818909121118371843799592960,28472613161534902071627567919297331348486838233018341
%N A381373 Sum over all partitions of [n] of n^j for a partition with j inversions.
%H A381373 Alois P. Heinz, <a href="/A381373/b381373.txt">Table of n, a(n) for n = 0..41</a>
%H A381373 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion</a>
%H A381373 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A381373 a(n) = Sum_{j>=0} n^j * A125810(n,j).
%F A381373 a(n) = A381369(n,n).
%F A381373 a(n) mod n = A062173(n) for n>=1.
%F A381373 a(n) mod 2 = A120325(n+1) for n>=1.
%p A381373 b:= proc(o, u, t, k) option remember;
%p A381373      `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2, k), 0)+add(k^(u+j-1)*
%p A381373         b(o-j, u+j-1, min(2, t+1), k), j=`if`(t=0, 1, 1..o)))
%p A381373     end:
%p A381373 a:= n-> b(n, 0$2, n):
%p A381373 seq(a(n), n=0..15);
%t A381373 b[o_, u_, t_, k_] := b[o, u, t, k] =
%t A381373    If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0, k], 0] + Sum[k^(u + j - 1)*
%t A381373    b[o - j, u + j - 1, Min[2, t + 1], k], {j, If[t == 0, {1}, Range[o]]}]];
%t A381373 a[n_] := b[n, 0, 0, n];
%t A381373 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Mar 15 2025, after _Alois P. Heinz_ *)
%Y A381373 Main diagonal of A381369.
%Y A381373 Cf. A062173, A120325, A125810, A381427.
%K A381373 nonn
%O A381373 0,3
%A A381373 _Alois P. Heinz_, Feb 21 2025