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A159749 The decomposition of a certain labeled universe (A052584), triangle read by rows. 1
2, 2, 4, 2, 12, 16, 0, 24, 96, 96, -8, 0, 320, 960, 768, 0, -240, 0, 4800, 11520, 7680, 240, 0, -6720, 0, 80640, 161280, 92160, 0, 13440, 0, -188160, 0, 1505280, 2580480, 1290240, -24192, 0, 645120, 0, -5419008, 0, 30965760, 46448640, 20643840 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..44.

FORMULA

T(n,k) = (n+1)!*C(n,k)*B_{n-k}*2^(k+1)/(k+1).

T(n,n) = A066318(n+1) = n!*2^(n+1) (necklaces with n labeled beads of 2 colors; see also A032184).

Sum_{k=0..n} T(n,k) = A052584(n+1) = (n+1)!*(1+2^n).

EXAMPLE

2

2, 4

2, 12, 16

0, 24, 96, 96

-8, 0, 320, 960, 768

0, -240, 0, 4800, 11520, 7680

240, 0, -6720, 0, 80640, 161280, 92160

MAPLE

T := (n, k) -> (n+1)!*binomial(n, k)*bernoulli(n-k, 1)*2^(k+1)/(k+1);

MATHEMATICA

T[n_, k_] := (n+1)! Binomial[n, k] BernoulliB[n-k, 1] 2^(k+1)/(k+1);

Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-Fran├žois Alcover, Jun 17 2019 *)

CROSSREFS

Cf. A027641, A027642, A052584.

Sequence in context: A059427 A137777 A126984 * A227293 A102416 A227509

Adjacent sequences:  A159746 A159747 A159748 * A159750 A159751 A159752

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Apr 20 2009

STATUS

approved

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Last modified December 5 17:42 EST 2019. Contains 329768 sequences. (Running on oeis4.)