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A285865
Denominator of the coefficient triangle B2(n, m) of generalized Bernoulli polynomials PB2(n, x) read by rows.
5
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.
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
KEYWORD
nonn,easy,tabl,frac
AUTHOR
Wolfdieter Lang, May 03 2017
STATUS
approved