A219976 Denominators of the Inverse bi-binomial transform of A164558(n)/A027642(n) read downwards antidiagonals.

1, 2, 2, 6, 6, 6, 1, 3, 3, 1, 30, 30, 30, 30, 30, 1, 15, 15, 15, 15, 1, 42, 42, 210, 210, 210, 42, 42, 1, 21, 21, 105, 105, 21, 21, 1, 30, 30, 210, 210, 210, 210, 210, 30, 30, 1, 15, 15, 105, 105, 105, 105, 15, 15, 1

Offset

0,2

Comments

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).

Links

Table of n, a(n) for n=0..54.

Example

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;

Mathematica

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 *)

Crossrefs

Cf. A213268.

Sequence in context: A167909 A368745 A185421 * A063944 A086447 A324032

Adjacent sequences: A219973 A219974 A219975 * A219977 A219978 A219979

Keyword

nonn,tabl,less

Author

Paul Curtz, Dec 02 2012

Status

approved