|
Uses the general transformation proposed by P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44 (a special case was studied by P. Flajolet, X. Gourdon and B. Salvy, idem, 1993, no. 55, pp. 67-78).
P(n,0)=u(0) P(n,x)=u(n)+x*sum[u(i)*P(n-i+1),x] i=0..n-1 for Bernoulli numbers with zeros, we obtain the fractional array and the inverse:
1 1
-1/2 1 1/2 1
1/6 -1 1 (1) 1/3 1 1 (2)
0 7/12 -3/2 1 5/24 11/12 3/2 1
The first column terms of (2) are:
1 1/2 1/3 5/24 7/60 49/720 43/840
Multiplied by n! n=1.. they give the sequence (1992,my exercice book p. 18).
Multiplied by n! n=1.. (1) and (2) become
1 1
-1 2 1 2
1 -6 6 (3) 2 6 6 (4)
0 14 -36 24 5 22 36 24
-4 -20 150 -240 120 14 90 210 240 120
Row sums of (3) are: Differences:
1 1 1 2 6 2 -6 1,3,13,74
Diagonals are interesting too.
| 