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A159749 The decomposition of a certain labeled universe (A052584), triangle read by rows. 1

%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-Fran├žois Alcover_, Jun 17 2019 *)

%Y Cf. A027641, A027642, A052584.

%K sign,tabl

%O 0,1

%A _Peter Luschny_, Apr 20 2009

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Last modified April 21 04:34 EDT 2021. Contains 343146 sequences. (Running on oeis4.)