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

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

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-François Alcover, Jun 17 2019 *)

Crossrefs

Cf. A027641, A027642, A052584.

Sequence in context: A059427 A137777 A126984 * A227293 A331391 A359442

Adjacent sequences: A159746 A159747 A159748 * A159750 A159751 A159752

Keyword

sign,tabl

Author

Peter Luschny, Apr 20 2009

Status

approved