%I #39 Sep 08 2022 08:46:00
%S 0,1,10,73,490,3246,21814,150535,1072786,7915081,60512348,479371384,
%T 3932969516,33392961185,293143783762,2658128519225,24872012040510,
%U 239916007100054,2383444110867378,24363881751014383,256034413642582418,2763708806499744097
%N Sum of dimension exponents of supercharacter of unipotent upper triangular matrices.
%C Supercharacter theory of unipotent upper triangular matrices over a finite field F(2) is indexed by set partitions S(n) of {1,2,..., n} where a set partition P of {1,2,..., n} is a subset { (i,j) : 1 <= i < j <= n}
%C such that (i,j) in P implies (i,k),(k,j) are not in P for all i<l<j.
%C The dimension of the representation associated to the supercharacter indexed by P is given by 2^Dim(P) where Dim(P) = sum [ j-i , (i,j) in P ].
%C The sequence we have is a(n) = sum [ Dim(P) , P in S(n) ].
%H Vincenzo Librandi, <a href="/A200580/b200580.txt">Table of n, a(n) for n = 1..200</a>
%H M. Aguiar, C. Andre, C. Benedetti, N. Bergeron, Z. Chen, P. Diaconis, A. Hendrickson, S. Hsiao, I.M. Isaacs, A. Jedwab, K. Johnson, G. Karaali, A. Lauve, T. Le, S. Lewis, H. Li, K. Magaard, E. Marberg, J-C. Novelli, A. Pang, F. Saliola, L. Tevlin, J-Y. Thibon, N. Thiem, V. Venkateswaran, C.R. Vinroot, N. Yan and M. Zabrocki, <a href="http://arxiv.org/abs/1009.4134">Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras</a>, arXiv:1009.4134 [math.CO], 2010-2011.
%H C. André, <a href="https://doi.org/10.1006/jabr.2001.8734">Basic characters of the unitriangular group</a>, Journal of Algebra, 175 (1995), 287-319.
%H B. Chern, P. Diaconis, D. M. Kane and R. C. Rhoades, <a href="http://math.stanford.edu/~rhoades/FILES/setpartitions.pdf">Closed expressions for averages of set partition statistics</a>, 2013.
%H Mikhail Khovanov, Victor Ostrik and Yakov Kononov, <a href="https://arxiv.org/abs/2011.14758">Two-dimensional topological theories, rational functions and their tensor envelopes</a>, arXiv:2011.14758 [math.QA], 2020.
%F a(n) = -2*B(n+2) + (n+4)*B(n+1) where B(i) = Bell numbers A000110. [Chern et al.] - _N. J. A. Sloane_, Jun 10 2013 [for offset 2]
%F a(n) ~ n^3 * Bell(n) / LambertW(n)^2 * (1 - 2/LambertW(n)). - _Vaclav Kotesovec_, Jul 28 2021
%p b:=proc(n,k) option remember;
%p if n=1 and k=1 then RETURN(1) fi;
%p if k=1 then RETURN(b(n-1,n-1)) fi;
%p b(n,k-1)+b(n-1,k-1)
%p end:
%p a:=proc(n) local res,k;
%p res:=0;
%p for k to n-1 do res:=res+k*(n-k)*b(n,k) od;
%p res
%p end:
%p seq(a(n),n=1..34);
%t Table[-2 BellB[n+3] + (n+5) BellB[n+2], {n, 1, 30}] (* _Vincenzo Librandi_, Jul 16 2013 *)
%o (Magma) [-2*Bell(n+3)+(n+5)*Bell(n+2): n in [1..30]]; // _Vincenzo Librandi_, Jul 16 2013
%Y Cf. A011971 (sequence is computed from the Aitken's array b(n,k)
%Y a(n) = sum [ k*(n-k)*b(n,k), k=1..n-1 ]).
%Y Cf. A200660, A200673 (other statistics related to supercharacter theory).
%Y Cf. A000110, A226507.
%K nonn
%O 1,3
%A _Nantel Bergeron_, Nov 19 2011