A285865 Denominator of the coefficient triangle B2(n, m) of generalized Bernoulli polynomials PB2(n, x) read by rows.

1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 15, 1, 3, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 33, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

0,4

Comments

The numerator triangle is given in A285864, where details are given.

Links

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

Formula

a(n, m) = denominator(B2(n, m)) with B2(n, m) = binomial(m, m)*2^(n-m)*B(n-m), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).

E.g.f. of the rational column sequences {B2(n, m)}_{n>=0} is 2*x/(exp(2*x) - 1)*x^m/m!. Here a(n, m) are the denominators of the exponentially generated sequence.

Example

The triangle a(n, m) begins:
n\m   0 1 2 3 4 5 6 7 8 9 10 ...
0:    1
1:    1 1
2:    3 1 1
3:    1 1 1 1
4:   15 1 1 1 1
5:    1 3 1 3 1 1
6:   21 1 1 1 1 1 1
7:    1 3 1 3 1 1 1 1
8:   15 1 3 1 3 1 3 1 1
9:    1 5 1 1 1 5 1 1 1 1
10:  33 1 1 1 1 1 1 1 1 1  1
...
For the triangle of the rationals B2(n, m) see A285864.

Mathematica

T[n_, m_]:=Denominator[Binomial[n, m]*2^(n - m)*BernoulliB[n - m]]; Table[T[n, m], {n, 0, 20}, {m, 0, n}] // Flatten (* Indranil Ghosh, May 06 2017 *)

Prog

(PARI) T(n, m) = denominator(binomial(n, m)*2^(n - m)*bernfrac(n - m));
for(n=0, 20, for(m=0, n, print1(T(n, m), ", "); ); print(); ) \\ Indranil Ghosh, May 06 2017
(Python)
from sympy import binomial, bernoulli
def T(n, m):
    return (binomial(n, m) * 2**(n - m) * bernoulli(n - m)).denominator()
for n in range(21): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, May 06 2017

Crossrefs

Cf. A027641/A027642, A285864.

Sequence in context: A204116 A093421 A146531 * A360098 A318451 A030381

Adjacent sequences: A285862 A285863 A285864 * A285866 A285867 A285868

Keyword

nonn,easy,tabl,frac

Author

Wolfdieter Lang, May 03 2017

Status

approved