OFFSET
0,2
COMMENTS
Difference table of Bernoulli numbers with B(1)=2/3:
1, 2/3, 1/6, 0, -1/30, 0, 1/42, 0, ...
-1/3, -1/2, -1/6, -1/30, 1/30, 1/42, -1/42, ...
-1/6, 1/3, 2/15, 1/15, -1/105, -1/21, ...
1/2, -1/5, -1/15, -8/105, -4/105, ...
-7/10, 2/15, -1/105, 4/105, ...
5/6, -1/7, 1/21, ...
-41/42, 2/15, ...
7/6, ...
...
First column: 1, -1/3, -1/6, 1/2, -7/10, 5/6, -41/42, 7/6, -41/30, 3/2, -35/22, 11/6, ... . a(n) is the n-th term of the denominators.
Antidiagonal sums: 1, 1/3, -1/2, 2/3, -5/6, 1, -7/6, 4/3, -3/2, 5/3, -11/6, 2, ... . See A060789(n).
a(2n+2)/a(2n+1) = 2, 5, 7, 5, 11, 455, ... .
By definition, for B(1) = b, the inverse binomial transform is
Bi(b) = 1, -1 + b, 7/6 - 2*b, -3/2 + 3*b, 59/30 + 4*b, ...
With Bic(b) = 0, -1/2 + b, 1 - 2*b, -3/2 + 3*b, 2 + 4*b, ...
= (-1)^n *(A001477(n)/2 - n*b),
FORMULA
EXAMPLE
a(0) = 1-0, a(1) = -1/2 +1/6 = -1/3, a(2) = 1/6 -1/3 = -1/6, a(3) = 0 +1/2.
MATHEMATICA
max = 66; B[1] = 2/3; B[n_] := BernoulliB[n]; BB = Array[B, max, 0]; a[n_] := Differences[BB, n] // First // Denominator; Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, May 11 2015 *)
PROG
(Sage)
def A257106_list(len, B1) :
T = matrix(QQ, 2*len+1)
for m in (0..2*len) :
T[0, m] = bernoulli_polynomial(1, m) if m <> 1 else B1
for k in range(m-1, -1, -1) :
T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
return [denominator(T[k, 0]) for k in (0..len-1)]
A257106_list(66, 2/3) # Peter Luschny, May 09 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Curtz, Apr 23 2015
STATUS
approved