The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A100615 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). 8
 1, -1, 5, -1, 1, 1, -5, -1, 7, 3, -15, -5, 7601, 691, -91, -35, 3617, 3617, -745739, -43867, 3317609, 1222277, -5981591, -854513, 5436374093, 1181820455, -213827575, -76977927, 213745149261, 23749461029, -249859397004145, -8615841276005, 238988952277727, 84802531453387 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 REFERENCES 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)}.] LINKS Robert Israel, Table of n, a(n) for n = 0..575 Madeline Beals-Reid, A Quadratic Relation in the Bernoulli Numbers, The Pump Journal of Undergraduate Research, 6 (2023), 29-39. FORMULA E.g.f.: (x/(exp(x)-1))^2. - Vladeta Jovovic, Feb 27 2006 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 a(n) = numerator(Sum_{j=0..n} binomial(n,j)*Bernoulli(n-j)*Bernoulli(j)). - Fabián Pereyra, Mar 02 2020 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 EXAMPLE 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. MAPLE S:= series((x/(exp(x)-1))^2, x, 41): seq(numer(coeff(S, x, j)*j!), j=0..40); # Robert Israel, Jun 02 2015 # Second program: a := n -> if n = 0 then 1 else numer(-n*bernoulli(n-1) - (n-1)*bernoulli(n)) fi: seq(a(n), n = 0..33); # Peter Luschny, May 18 2023 MATHEMATICA Table[Numerator@NorlundB[n, 2], {n, 0, 32}] (* Arkadiusz Wesolowski, Oct 22 2012 *) Table[If[n == 0, 1, -Numerator[n*BernoulliB[n - 1] + (n - 1)*BernoulliB[n]]], {n, 0, 33}] (* Peter Luschny, May 18 2023 *) PROG (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 */ (PARI) a(n) = numerator(sum(j=0, n, binomial(n, j)*bernfrac(n-j)*bernfrac(j))); \\ Michel Marcus, Mar 03 2020 CROSSREFS Cf. A001898, A027641, A027642, A100616, A359738. Sequence in context: A066803 A089608 A250131 * A293897 A334988 A334987 Adjacent sequences: A100612 A100613 A100614 * A100616 A100617 A100618 KEYWORD sign,frac AUTHOR N. J. A. Sloane, Dec 03 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 6 03:00 EST 2023. Contains 367594 sequences. (Running on oeis4.)