A100615 - OEIS

%I #39 May 18 2023 17:30:07

%S 1,-1,5,-1,1,1,-5,-1,7,3,-15,-5,7601,691,-91,-35,3617,3617,-745739,

%T -43867,3317609,1222277,-5981591,-854513,5436374093,1181820455,

%U -213827575,-76977927,213745149261,23749461029,-249859397004145,-8615841276005,238988952277727,84802531453387

%N Let N(n)(x) be the Nørlund polynomials as defined in A001898, with N(n)(1) equal to the usual Bernoulli numbers A027641/A027642. Sequence gives numerators of N(n)(2).

%C With the signs of A359738, the rational sequence reflects the identity B(z)^2 = (z + 1)*B(z) - z*B'(z), that goes back to Euler, where B(z) = z/(1 - e^(-z)) is the e.g.f. of the Bernoulli numbers with B(1) = 1/2. - _Peter Luschny_, Jan 23 2023

%D F. N. David, Probability Theory for Statistical Methods, Cambridge, 1949; see pp. 103-104. [There is an error in the recurrence for B_s^{(r)}.]

%H Robert Israel, <a href="/A100615/b100615.txt">Table of n, a(n) for n = 0..575</a>

%H Madeline Beals-Reid, <a href="https://journals.calstate.edu/pump/article/view/3549">A Quadratic Relation in the Bernoulli Numbers</a>, The Pump Journal of Undergraduate Research, 6 (2023), 29-39.

%F E.g.f.: (x/(exp(x)-1))^2. - _Vladeta Jovovic_, Feb 27 2006

%F a(n) = numerator(Sum_{k=0..n}(-1)^k*k!/(k+1)*Sum_{j=0..n-k} C(n,j)*Stirling2(n-j,k)*B(j)), where B(n) is Bernoulli numbers. - _Vladimir Kruchinin_, Jun 02 2015

%F a(n) = numerator(Sum_{j=0..n} binomial(n,j)*Bernoulli(n-j)*Bernoulli(j)). - _Fabián Pereyra_, Mar 02 2020

%F a(n) = -numerator(n*B(n-1) + (n-1)*B(n)) for n >= 1, where B(n) = Bernoulli(n, 0). - _Peter Luschny_, May 18 2023

%e 1, -1, 5/6, -1/2, 1/10, 1/6, -5/42, -1/6, 7/30, 3/10, -15/22, -5/6, 7601/2730, 691/210, -91/6, -35/2, 3617/34, 3617/30, -745739/798, -43867/42, ... = A100615/A100616.

%p S:= series((x/(exp(x)-1))^2, x, 41):

%p seq(numer(coeff(S,x,j)*j!), j=0..40); # _Robert Israel_, Jun 02 2015

%p # Second program:

%p a := n -> if n = 0 then 1 else numer(-n*bernoulli(n-1) - (n-1)*bernoulli(n)) fi:

%p seq(a(n), n = 0..33);  # _Peter Luschny_, May 18 2023

%t Table[Numerator@NorlundB[n, 2], {n, 0, 32}] (* _Arkadiusz Wesolowski_, Oct 22 2012 *)

%t Table[If[n == 0, 1, -Numerator[n*BernoulliB[n - 1] + (n - 1)*BernoulliB[n]]], {n, 0,  33}] (* _Peter Luschny_, May 18 2023 *)

%o (Maxima) a(n):=sum((-1)^k*k!/(k+1)*sum(binomial(n,j)*stirling2(n-j,k)*bern(j),j,0,n-k),k,0,n); /* _Vladimir Kruchinin_, Jun 02 2015 */

%o (PARI) a(n) = numerator(sum(j=0, n, binomial(n,j)*bernfrac(n-j)*bernfrac(j))); \\ _Michel Marcus_, Mar 03 2020

%Y Cf. A001898, A027641, A027642, A100616, A359738.

%K sign,frac

%O 0,3

%A _N. J. A. Sloane_, Dec 03 2004