|
%I
%S 2,2,4,2,12,16,0,24,96,96,-8,0,320,960,768,0,-240,0,4800,11520,7680,
%T 240,0,-6720,0,80640,161280,92160,0,13440,0,-188160,0,1505280,2580480,
%U 1290240,-24192,0,645120,0,-5419008,0,30965760,46448640,20643840
%N The decomposition of a certain labeled universe (A052584), triangle read by rows.
%C T(n,k) is a weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1) = 1, which amounts to the definition B_{n} = B_{n}(1).
%F T(n,k) = (n+1)!*C(n,k)*B_{n-k}*2^(k+1)/(k+1).
%F T(n,n) = A066318(n+1) = n!*2^(n+1) (necklaces with n labeled beads of 2 colors; see also A032184).
%F Sum_{k=0..n} T(n,k) = A052584(n+1) = (n+1)!*(1+2^n).
%e 2
%e 2, 4
%e 2, 12, 16
%e 0, 24, 96, 96
%e -8, 0, 320, 960, 768
%e 0, -240, 0, 4800, 11520, 7680
%e 240, 0, -6720, 0, 80640, 161280, 92160
%p T := (n,k) -> (n+1)!*binomial(n,k)*bernoulli(n-k,1)*2^(k+1)/(k+1);
%t T[n_, k_] := (n+1)! Binomial[n, k] BernoulliB[n-k, 1] 2^(k+1)/(k+1);
%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* _Jean-François Alcover_, Jun 17 2019 *)
%Y Cf. A027641, A027642, A052584.
%K sign,tabl
%O 0,1
%A _Peter Luschny_, Apr 20 2009
|
