0,2

Starting from any sequence a(k) in the first row, we define the array T(n,k) of the inverse bi-binomial transform by T(0,k) = a(k), T(n,k) = T(n-1,k+1) -2*T(n-1,k) n>0. Hence A164558(n)/A027642(n) and successive "bi-differences":

1, 3/2, 13/6, 3, 119/30, 5, 253/42, 7, 239/30, 9;

-1/2, -5/6, -4/3, -61/30, -44/15, -167/42, -106/21, -181/30, -104/15;

1/6, 1/3, 19/30, 17/15, 397/210, 61/21 , 853/210, 77/15;

0, -1/30, -2/15, -79/210, -92/105, -367/210, -314/105;

-1/30, -1/15, -23/210, -13/105, 1/210, 53/105;

0, 1/42, 2/21, 53/210, 52/105;

1/42, 1/21, 13/210, -1/105;

0, -1/30, -2/15;

-1/30, -1/15;

0.

The first column is A027641(n)/A027642(n).

Partial array of denominators:

1, 2, 6, 1, 30, 1, 42, 1, 30, 1;

2, 6, 3, 30, 15, 42, 21, 30, 15;

6, 3, 30, 15, 210, 21, 210, 15;

1, 30, 15, 210, 105, 210, 105;

30, 15, 210, 105, 210, 105;

1, 42, 21, 210, 105;

42, 21, 210, 105;

1, 30, 15;

30, 15;

1.

a(n):

1;

2, 2;

6, 6, 6,;

1, 3, 3, 1;

30, 30, 30, 30, 30;

A164558[n_] := Sum[(-1)^k*Binomial[n, k]*BernoulliB[k], {k, 0, n}] // Numerator; t[0, k_?Positive] := A164558[k] / Denominator[ BernoulliB[k]]; t[n_?Positive, k_] := t[n, k] = t[n-1, k+1] - 2*t[n-1, k]; t[0, 0] = 1; t[_, _] = 0; Flatten[ Table[t[n-k , k] // Denominator, {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Dec 04 2012 *)

Cf. A213268.

Sequence in context: A167909 A368745 A185421 * A063944 A086447 A324032

Adjacent sequences: A219973 A219974 A219975 * A219977 A219978 A219979

nonn,tabl,less

Paul Curtz, Dec 02 2012

approved