cp's OEIS Frontend

A381427 Sum over all ordered partitions of [n] of n^j for an ordered partition with j inversions.

%I A381427 #14 Feb 23 2025 17:34:42
%S A381427 1,1,4,79,14808,40065301,2099255895008,2651651342949844915,
%T A381427 96254339565438079064819328,116387990444553949414146511586296381,
%U A381427 5327195120249449992420082364255283659438679552,10333056290045508772052838892223597279253890797441054043823
%N A381427 Sum over all ordered partitions of [n] of n^j for an ordered partition with j inversions.
%H A381427 Alois P. Heinz, <a href="/A381427/b381427.txt">Table of n, a(n) for n = 0..36</a>
%H A381427 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion</a>
%H A381427 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A381427 a(n) = Sum_{j=0..binomial(n,2)} n^j * A381299(n,j).
%F A381427 a(n) = A381426(n,n).
%F A381427 a(n) mod n = A062173(n) for n>=1.
%F A381427 a(n) mod 2 = A135528(n+1).
%p A381427 b:= proc(o, u, t, k) option remember; `if`(u+o=0, 1, `if`(t=1,
%p A381427       b(u+o, 0$2, k), 0)+add(k^(u+j-1)*b(o-j, u+j-1, 1, k), j=1..o))
%p A381427     end:
%p A381427 a:= n-> b(n, 0$2, n):
%p A381427 seq(a(n), n=0..11);
%Y A381427 Main diagonal of A381426.
%Y A381427 Cf. A062173, A135528, A381299, A381373.
%K A381427 nonn
%O A381427 0,3
%A A381427 _Alois P. Heinz_, Feb 23 2025