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A257106 Denominators of the inverse binomial transform of the Bernoulli numbers with B(1)=2/3. 1
1, 3, 6, 2, 10, 6, 42, 6, 30, 2, 22, 6, 2730, 6, 6, 2, 170, 6, 798, 6, 330, 2, 46, 6, 2730, 6, 6, 2, 290, 6, 14322, 6, 510, 2, 2, 6, 1919190, 6, 6, 2, 4510, 6, 1806, 6, 690, 2, 94, 6, 46410, 6, 66, 2, 530, 6, 798, 6, 870, 2, 118, 6, 56786730, 6, 6, 2, 170, 6 (list; graph; refs; listen; history; text; internal format)
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, ...

      = A176328(n)/A176591(n) - (-1)^n *n*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),

Bi(b) = (-1)^n *(A164555(n)/A027642(n) + A001477(n)/2 - n*b) =

      = A027641(n)/A027642(n) + Bic(b) .

LINKS

Table of n, a(n) for n=0..65.

FORMULA

Conjecture: a(2n+1) = 3 followed by period 3: repeat 2, 6, 6.

Conjecture: a(2n) = A002445(n)/(period 3: repeat 1, 1, 3).

a(n) = A027641(n)/A027642(n) - (-1)^n *n/6.

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-Fran├žois 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

Cf. A256595, A027641/A027642(n), A164555(n)/A027642(n), A060789, A176328/A176591, A001477, A109007.

Sequence in context: A154204 A309609 A266971 * A210187 A210189 A257506

Adjacent sequences:  A257103 A257104 A257105 * A257107 A257108 A257109

KEYWORD

nonn,changed

AUTHOR

Paul Curtz, Apr 23 2015

STATUS

approved

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Last modified November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)