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

%I A285865 #18 Jun 27 2025 15:30:01
%S A285865 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,
%T A285865 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
%N A285865 Denominator of the coefficient triangle B2(n, m) of generalized Bernoulli polynomials PB2(n, x) read by rows.
%C A285865 The numerator triangle is given in A285864, where details are given.
%F A285865 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).
%F A285865 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.
%e A285865 The triangle a(n, m) begins:
%e A285865 n\m   0 1 2 3 4 5 6 7 8 9 10 ...
%e A285865 0:    1
%e A285865 1:    1 1
%e A285865 2:    3 1 1
%e A285865 3:    1 1 1 1
%e A285865 4:   15 1 1 1 1
%e A285865 5:    1 3 1 3 1 1
%e A285865 6:   21 1 1 1 1 1 1
%e A285865 7:    1 3 1 3 1 1 1 1
%e A285865 8:   15 1 3 1 3 1 3 1 1
%e A285865 9:    1 5 1 1 1 5 1 1 1 1
%e A285865 10:  33 1 1 1 1 1 1 1 1 1  1
%e A285865 ...
%e A285865 For the triangle of the rationals B2(n, m) see A285864.
%t A285865 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 *)
%o A285865 (PARI) T(n, m) = denominator(binomial(n, m)*2^(n - m)*bernfrac(n - m));
%o A285865 for(n=0, 20, for(m=0, n, print1(T(n, m),", ");); print();) \\ _Indranil Ghosh_, May 06 2017
%o A285865 (Python)
%o A285865 from sympy import binomial, bernoulli
%o A285865 def T(n, m):
%o A285865     return (binomial(n, m) * 2**(n - m) * bernoulli(n - m)).denominator
%o A285865 for n in range(21): print([T(n, m) for m in range(n + 1)]) # _Indranil Ghosh_, May 06 2017
%Y A285865 Cf. A027641/A027642, A285864.
%K A285865 nonn,easy,tabl,frac
%O A285865 0,4
%A A285865 _Wolfdieter Lang_, May 03 2017