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%I #46 Oct 29 2020 15:09:16
%S 1,1,2,4,14,46,282,1394,12658,83122,985730,8012962,116597538,
%T 1127575970,19410377378,217492266658,4320408974978,55023200887938,
%U 1238467679662722,17665859065690754,444247724347355554,7015393325151055906,194912434760367113570,3375509056735963889634
%N Number of permutations of degree n with barycenter 0.
%C The barycenter or signcenter of a permutation is the sum of the signs of the difference between initial and final positions of the objects.
%H Alois P. Heinz, <a href="/A062868/b062868.txt">Table of n, a(n) for n = 0..450</a> (first 151 terms from Maxwell Jiang)
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, n-2*k)*A320337(k). - _Maxwell Jiang_, Dec 19 2018 (added by editors)
%F a(n) ~ sqrt(3) * (1 + exp(-2)*(-1)^n) * n^n / exp(n). - _Vaclav Kotesovec_, Oct 29 2020
%e (4,1,3,5,2) has difference (3,-1,0,1,-3) and signs (1,-1,0,1,-1) with total 0.
%p b:= proc(s, t) option remember; (n-> `if`(abs(t)>n, 0, `if`(n=0, 1,
%p add(b(s minus {j}, t+signum(n-j)), j=s))))(nops(s))
%p end:
%p a:= n-> b({$1..n}, 0):
%p seq(a(n), n=0..14); # _Alois P. Heinz_, Jul 31 2018
%t E1[n_ /; n >= 0, 0] = 1;
%t E1[n_, k_] /; k < 0 || k > n = 0;
%t E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
%t b[n_] := Sum[(-1)^(n-k) E1[n+k, n] Binomial[2n, n-k], {k, 0, n}];
%t a[n_] := Sum[Binomial[n, n-2k] b[k], {k, 0, n/2}];
%t a /@ Range[0, 150] (* _Jean-François Alcover_, Oct 29 2020, after _Peter Luschny_ in A320337 *)
%Y Column k=0 of A062866 or of A062867.
%Y Cf. A179567, A196687, A196688, A320337.
%K nice,nonn
%O 0,3
%A _Olivier Gérard_, Jun 26 2001
%E One more term from _Vladeta Jovovic_, Jun 28 2001
%E a(11)-a(14) from _Hugo Pfoertner_, Sep 23 2004
%E a(15)-a(18) from _R. H. Hardin_, Jul 18 2010
%E a(19)-a(22) from _Kyle G Hess_, Jul 30 2018
%E a(0)=1 prepended by _Alois P. Heinz_, Jul 30 2018
%E Terms a(23) and beyond from _Maxwell Jiang_, Dec 19 2018
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