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

Author

Wolfdieter Lang, May 03 2017

Keywords

• nonn
• easy
• tabl
• frac

Comments

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

Examples

			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.
		

Crossrefs

Cf. A027641/A027642, A285864.

Programs

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 *)

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

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.