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

Author

Paul Curtz, Dec 02 2012

Keywords

• nonn
• tabl
• less

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).

Examples

			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;
		

Crossrefs

Cf. A213268.

Programs

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 *)