cp's OEIS Frontend

A164558 Numerators of the n-th term of the binomial transform of the original Bernoulli numbers.

%I A164558 #42 Feb 24 2023 07:16:15
%S A164558 1,3,13,3,119,5,253,7,239,9,665,11,32069,13,91,15,4543,17,58231,19,
%T A164558 -168011,21,857549,23,-236298571,25,8553259,27,-23749436669,29,
%U A164558 8615841705665,31,-7709321024897,33,2577687858571,35,-26315271552984386533,37,2929993913841787
%N A164558 Numerators of the n-th term of the binomial transform of the original Bernoulli numbers.
%C A164558 We start from the sequence A164555(i)/A027642(i) of the "original" Bernoulli numbers, i >= 0, and compute its binomial transform, which is the sequence of fractions 1, 3/2, 13/6, 3, 119/30, 5, 253/42, 7, 239/30, 9, ... The a(n) are the numerators of these fractions.
%C A164558 These fractions are also the successive values of Bernoulli(n,2). - _N. J. A. Sloane_, Nov 10 2009
%C A164558 (-1)^n*a(n)/A027642, with e.g.f. x/(exp(x)*(exp(x)-1)), gives the alternating row sums of the triangle of coefficients of the Bernoulli polynomials A053382/A053383 or A196838/A196839. - _Wolfdieter Lang_, Oct 25 2011
%H A164558 Alois P. Heinz, <a href="/A164558/b164558.txt">Table of n, a(n) for n = 0..250</a>
%F A164558 E.g.f. for a(n)/A027642: x/(exp(-x)*(1-exp(-x))). - _Wolfdieter Lang_, Oct 25 2011
%F A164558 Let b_{n}(x) = B_{n}(x) - 2*x*[x^(n-1)]B_{n}(x), then a(n) = numerator(b_{n}(1)). - _Peter Luschny_, Jun 15 2012
%F A164558 Numerators of the polynomials b(n,x) generated by exp(x*z)*z/(1-exp(-z)) evaluated x=1. b(n,x) are the Bernoulli polynomials B(n,x) with a different sign schema, b(n,x) = (-1)^n*B(n,-x) (see the example section). In other words: a(n) = numerator((-1)^n*Bernoulli(n,-1)). a(n) = n for odd n >= 3. - _Peter Luschny_, Aug 18 2018
%e A164558 Numerators of the polynomials b(n,x) at x=1 for n >= 0. The first few are: 1, 1/2 + x, 1/6 + x + x^2, (1/2)*x + (3/2)*x^2 + x^3, -1/30 + x^2 + 2*x^3 + x^4, -(1/6)*x +(5/3)*x^3 + (5/2)*x^4 + x^5, ... - _Peter Luschny_, Aug 18 2018
%p A164558 read("transforms") : nmax := 40: a := BINOMIAL([seq(A164555(n)/A027642(n),n=0..nmax)]) : seq( numer(op(n,a)),n=1..nmax+1) ; # _R. J. Mathar_, Aug 26 2009
%p A164558 A164558 := n -> `if`(type(n, odd) and n > 1, n, numer((-1)^n*bernoulli(n,-1))):
%p A164558 seq(A164558(n), n=0..50); # _Peter Luschny_, Jun 15 2012, revised Aug 18 2018
%t A164558 a[n_]:= Sum[(-1)^k*Binomial[n, k]*BernoulliB[k], {k,0,n}]//Numerator;
%t A164558 Table[a[n], {n,0,50}] (* _Jean-François Alcover_, Aug 08 2012 *)
%o A164558 (PARI) a(n) = numerator(subst(bernpol(n, x), x, 2)); \\ _Michel Marcus_, Mar 03 2020
%o A164558 (Magma)
%o A164558 A164558:= func< n | Numerator((&+[(-1)^j*Binomial(n,j)*Bernoulli(j): j in [0..n]])) >;
%o A164558 [A164558(n): n in [0..50]]; // _G. C. Greubel_, Feb 24 2023
%o A164558 (SageMath)
%o A164558 def A164558(n): return sum((-1)^j*binomial(n,j)*bernoulli(j) for j in range(n+1)).numerator()
%o A164558 [A164558(n) for n in range(51)] # _G. C. Greubel_, Feb 24 2023
%Y A164558 Cf. A027642, A164555, A196838, A196839.
%K A164558 sign,frac
%O A164558 0,2
%A A164558 _Paul Curtz_, Aug 16 2009
%E A164558 Edited and extended by _R. J. Mathar_, Aug 26 2009