A177690 Denominators of the Inverse Akiyama-Tanigawa transform of the aerated even-indexed Bernoulli numbers 1, 0, 1/6, 0, -1/30, 0, 1/42, ...

1, 1, 12, 6, 120, 120, 280, 140, 5040, 5040, 55440, 55440, 720720, 720720, 720720, 360360, 24504480, 24504480, 155195040, 155195040, 31039008, 10346336, 237965728, 713897184, 17847429600, 17847429600, 160626866400, 22946695200

Offset: 0

Author

Paul Curtz, May 11 2010

Keywords

• nonn
• frac

Comments

See A177427 (numerators) for the description of the Akiyama-Tanigawa array of this sequence of fractions, T(0,k) = 1, 1, 13/12, 7/6, 149/120, 157/120, ...

If we add a zero in front and construct an array A(n,k) with successive differences A(n,k) = A(n-1,k+1)-A(n-1,k), the array A(.,.) becomes

0, 1, 1, 13/12, 7/6, 149/120, 157/120, 383/280, 199/140, 7409/5040, ...

1, 0, 1/12, 1/12, 3/40, 1/15, 5/84, 3/56, 7/144, 2/45, 9/220, ...

-1, 1/12, 0, -1/120, -1/120, -1/140, -1/168, -5/1008, -1/240, -7/1980, ...

13/12, -1/12, -1/120, 0, 1/840, 1/840, 1/1008, 1/1260, 1/1584, ...

-7/6, 3/40, 1/120, 1/840, 0, -1/5040, -1/5040, -1/6160, -1/7920, ...

149/120, -1/15, -1/140, -1/840, -1/5040, 0, 1/27720, 1/27720, ...

-157/120, 5/84, 1/168, 1/1008, 1/5040, 1/27720, 0, -1/144144, -1/144144, ...

On the diagonal, A(n,n)=0. The left column A(n,0) = (-1)^(n+1)*A(0,k) is a signed variant of the top row, which means the sequence is some eigensequence under the inverse binomial transform (see A174341 for other examples). This eigen-feature would remain if the same number of top rows and left columns were removed from A(.,.).

Crossrefs

Cf. A177427 (numerators).

Programs

Maple

read("transforms3") ; [seq(bernoulli(2*n),n=0..20)] ; AERATE(%,1) ; AKIYAMATANIGAWAi(%) ; apply(denom,%) ; # R. J. Mathar, Jan 16 2011

Mathematica

t[n_, 0] := BernoulliB[n]; t[1, 0]=0; t[n_, k_] := t[n, k] = (t[n, k-1] + (k-1)*t[n, k-1] - t[n+1, k-1])/k; Table[t[0, k], {k, 0, 27}] // Denominator (* Jean-François Alcover, Aug 09 2012 *)