A001898 -id:A001898 - cp's OEIS frontend

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).

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

Offset: 0

N. J. A. Sloane, Dec 03 2004

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

			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.

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)}.]

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.

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

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

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 *)

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 */

a(n) = numerator(sum(j=0, n, binomial(n,j)*bernfrac(n-j)*bernfrac(j))); \\ Michel Marcus, Mar 03 2020

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

A100616 Let B(n)(x) be the Bernoulli polynomials as defined in A001898, with B(n)(1) equal to the usual Bernoulli numbers A027641/A027642. Sequence gives denominators of B(n)(2).

Original entry on oeis.org

1, 1, 6, 2, 10, 6, 42, 6, 30, 10, 22, 6, 2730, 210, 6, 2, 34, 30, 798, 42, 330, 110, 46, 6, 2730, 546, 6, 2, 290, 30, 14322, 462, 510, 170, 2, 6, 54834, 51870, 6, 2, 4510, 330, 1806, 42, 690, 46, 94, 6, 46410, 6630, 66, 22, 530, 30, 798, 798, 174, 290, 118, 6, 56786730

Offset: 0

N. J. A. Sloane, Dec 03 2004

			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.

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)}.]

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A001898, A027641, A027642, A100615.

S:= series((x/(exp(x)-1))^2, x, 101):
seq(denom(coeff(S,x,n)*n!), n=0..100); # Robert Israel, Jun 02 2015

Table[Denominator@NorlundB[n, 2], {n, 0, 59}] (* Arkadiusz Wesolowski, Oct 22 2012 *)

a(n) = denominator(sum(j=0, n, binomial(n,j)*bernfrac(n-j)*bernfrac(j))); \\ Michel Marcus, Mar 03 2020

E.g.f.: (x/(exp(x)-1))^2. - Vladeta Jovovic, Feb 27 2006

a(n) = denominator(Sum_{j=0..n} binomial(n,j)*Bernoulli(n-j)*Bernoulli(j)). - Fabián Pereyra, Mar 02 2020

A100655 Triangle read by rows giving coefficients in Bernoulli polynomials as defined in A001898, after multiplication by the common denominators A001898(n).

Original entry on oeis.org

1, 0, -1, 0, -1, 3, 0, 0, 1, -1, 0, 2, 5, -30, 15, 0, 0, -2, -5, 10, -3, 0, -16, -42, 91, 315, -315, 63, 0, 0, 16, 42, -7, -105, 63, -9, 0, 144, 404, -540, -2345, -840, 3150, -1260, 135, 0, 0, -144, -404, -100, 665, 448, -630, 180, -15, 0, -768, -2288, 2068, 11792, 8195, -8085, -8778, 6930, -1485, 99

Offset: 0

N. J. A. Sloane, Dec 05 2004

Let p(n, x) = Sum_{k=0..n} T(n, k)*x^k, then the polynomials (-1)^n*p(n; x)/x are called 'Stirling polynomials' by Knuth et al. (CMath, eq. 6.45). - Peter Luschny, Feb 05 2021

			The Bernoulli polynomials B(0)(x) through B(6)(x) are:
        1
    -(1/2)* x
    (1/12)*(3*x - 1)*x
    -(1/8)*(x-1)*x^2
   (1/240)*(15*x^3 - 30*x^2 + 5*x + 2)*x
   -(1/96)*(x-1)*(3*x^2 - 7*x - 2)*x^2
  (1/4032)*(63*x^5 - 315*x^4 + 315*x^3 + 91*x^2 - 42*x - 16)*x
Triangle of coefficients starts:
[0] [1],
[1] [0,  -1],
[2] [0,  -1,   3],
[3] [0,   0,   1,   -1],
[4] [0,   2,   5,  -30,    15],
[5] [0,   0,  -2,   -5,    10,   -3],
[6] [0, -16, -42,   91,   315, -315,   63],
[7] [0,   0,  16,   42,    -7, -105,   63,    -9],
[8] [0, 144, 404, -540, -2345, -840, 3150, -1260, 135].

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 1st ed.; Addison-Wesley, 1989, p. 257.

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)}.]
N. E. Nörlund, Vorlesungen ueber Differenzenrechnung Springer 1924, (p. 146).

Cf. A001898, A027641, A027642, A053657, A100615, A100616, A201637.

CoeffList := p -> op(PolynomialTools:-CoefficientList(simplify(p),x)):
E2 := (n, k) -> combinat[eulerian2](n, k): m := n -> mul(j-x, j = 1..n):
Epoly := n -> simplify(expand(add(E2(n, k)*binomial(x+k,2*n), k = 0..n)/m(n))):
poly := n -> Epoly(n)*denom(Epoly(n)):
seq(print(CoeffList(poly(n))), n = 0..8); # Peter Luschny, Feb 05 2021

row[n_] := NorlundB[n, x] // Together // Numerator // CoefficientList[#, x]&; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 06 2019, after Peter Luschny *)

# Formula (83), page 146 in Nörlund.
@cached_function
def NoerlundB(n, x):
    if n == 0: return 1
    return expand((-x/n)*add((-1)^k*binomial(n,k)*bernoulli(k)*NoerlundB(n-k,x) for k in (1..n)))
def A100655_row(n): return numerator(NoerlundB(n, x)).list()
[A100655_row(n) for n in (0..8)] # Peter Luschny, Jul 01 2019

E.g.f.: (y/(exp(y)-1))^x. - Vladeta Jovovic, Feb 27 2006

Let p(n, x) = (Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n))/(Product_{j=1..n} (j-x)), where E2 are the second-order Eulerian numbers (A201637), then T(n, k) = [x^k] M(n+1)*p(n, x), where M(n) are the Minkowski numbers (A053657). - Peter Luschny, Feb 05 2021

A137524 Triangular sequence from coefficients of the umbral calculus expansion of a Golden -Mean Bernoulli function(A001898): p(x,t)=tphi^(xt)/(phi^t - 1), where the golden ratio replaces "e".

Original entry on oeis.org

2, -3, 6, 4, -24, 24, 0, 60, -180, 120, -24, 0, 720, -1440, 720, 0, -840, 0, 8400, -12600, 5040, 960, 0, -20160, 0, 100800, -120960, 40320, 0, 60480, 0, -423360, 0, 1270080, -1270080, 362880, -120960, 0, 2419200, 0, -8467200, 0, 16934400, -14515200, 3628800, 0, -11975040, 0, 79833600, 0, -167650560, 0

Offset: 1

Roger L. Bagula, Apr 27 2008

Row sums are: {2, 3, 4, 0, -24, 0, 960, 0, -120960, 0, 36288000}
These are the same as the Bernoulli numbers with the factor log(phi)^n: p[t_] = t*Exp[x*t]/(Exp[t] - 1);
a = Table[ CoefficientList[(n + 2)!*n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}].

			{2},
{-3, 6},
{4, -24, 24},
{0, 60, -180, 120},
{-24, 0, 720, -1440, 720},
{0, -840, 0, 8400, -12600, 5040},
{960, 0, -20160, 0, 100800, -120960, 40320},
{0, 60480, 0, -423360, 0, 1270080, -1270080, 362880},
{-120960, 0, 2419200, 0, -8467200, 0, 16934400, -14515200, 3628800},
{0, -11975040, 0, 79833600, 0, -167650560, 0, 239500800, -179625600, 39916800}, {36288000, 0, -718502400, 0, 2395008000, 0, -3353011200, 0, 3592512000, -2395008000, 479001600}

Cf. A001898.

p[t_]=t*GoldenRatio^(x*t)/(GoldenRatio^t-1); Table[ ExpandAll[((n+2)!*n!/Log[GoldenRatio]^(n-1))*SeriesCoefficient[ Series[p[t],{t,0,30}],n]],{n,0,10}]; a=Table[ CoefficientList[((n+2)!*n!/Log[GoldenRatio]^(n-1))*SeriesCoefficient[ Series[p[t],{t,0,30}],n],x],{n,0,10}]; Flatten[a] Table[Apply[Plus,CoefficientList[((n+2)!*n!/Log[GoldenRatio]^(n-1))*SeriesCoefficient[ Series[p[t],{t,0,30}],n],x]],{n,0,10}];

A053657 a(n) = Product_{p prime} p^{ Sum_{k>=0} floor[(n-1)/((p-1)p^k)]}.

Original entry on oeis.org

1, 2, 24, 48, 5760, 11520, 2903040, 5806080, 1393459200, 2786918400, 367873228800, 735746457600, 24103053950976000, 48206107901952000, 578473294823424000, 1156946589646848000, 9440684171518279680000, 18881368343036559360000, 271211974879377138647040000

Offset: 1

Jean-Luc Chabert, Feb 16 2000

LCM of denominators of the coefficients of x^n*z^k in {-log(1-x)/x}^z as k=0..n, as described by triangle A075264.
Denominators of integer-valued polynomials on prime numbers (with degree n): 1/a(n) is a generator of the ideal formed by the leading coefficients of integer-valued polynomials on prime numbers with degree less than or equal to n.
Also the least common multiple of the orders of all finite subgroups of GL_n(Q) [Minkowski]. Schur's notation for the sequence is M_n = a(n+1). - Martin Lorenz (lorenz(AT)math.temple.edu), May 18 2005
This sequence also occurs in algebraic topology where it gives the denominators of the Laurent polynomials forming a regular basis for K*K, the hopf algebroid of stable cooperations for complex K-theory. Several different equivalent formulas for the terms of the sequence occur in the literature. An early reference is K. Johnson, Illinois J. Math. 28(1), 1984, pp.57-63 where it occurs in lines 1-5, page 58. A summary of some of the other formulas is given in the appendix to K. Johnson, Jour. of K-theory 2(1), 2008, 123-145. - Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008
a(n) is divisible by n!, by Legendre's formula for the highest power of a prime that divides n!. Also, a(n) is divisible by (n+1)! if and only if n+1 is not prime. - Jonathan Sondow, Jul 23 2009
Triangle A163940 is related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m =>-1. The left hand columns of this triangle can be generated with the MC polynomials, see A163972. The Minkowski numbers appear in the denominators of these polynomials. - Johannes W. Meijer, Oct 16 2009
Unsigned Stirling numbers of the first kind as [s + k, k] (Karamata's notation) where k = {0, 1, 2, ...} and s is in general complex results in Pochhammer[s,k]*(integer coefficient polynomial of (k-1) degree in s) / M[k], where M[k] is the least common multiple of the orders of all finite groups of n x n-matrices over rational numbers (Minkowiski's theorem) which is sequence A053657. - Lorenz H. Menke, Jr., Feb 02 2010
From Peter Bala, Feb 21 2011: (Start)
Given a subset S of the integers Z, Bhargava has shown how to associate with S a generalized factorial function, denoted n!_S, which shares many properties of the classical factorial function n!.
The present sequence is the generalized factorial function n!S associated with the set of primes S = {2,3,5,7,...}. The associated generalized exponential function E(x) = Sum{n>=1} x^(n-1)/a(n) vanishes at x = -2: i.e. Sum_{n>=1} (-2)^n/a(n) = 0.
For the table of associated generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) see A186430.
This sequence is related to the Bernoulli polynomials in two ways [Chabert and Cahen]:
(1) a(n) = (n-1)!*A001898(n-1).
(2) (t/(exp(t)-1))^x = sum {n = 0..inf} P(n,x)*t^n/a(n+1),
where the P(n,x) are primitive polynomials in the ring Z[x].
If p_1,...,p_n are any n primes then the product of their pairwise differences Product_{i(End)
LCM of denominators of the coefficients of S(m+n-1,m) as polynomial in m of degree 2*(n-1), as described by triangle A202339. - Vladimir Shevelev, Dec 17 2011
Sometimes called "Minkowski numbers" (e.g., by Guralnick and Lorenz), after the German mathematician Hermann Minkowski (1864-1909). - Amiram Eldar, Aug 24 2024

			a(7)=24^3*Product_{i=1..3} A202318(i)=24^3*1*10*21=2903040. - _Vladimir Shevelev_, Dec 17 2011

Jean-Luc Chabert, Scott T. Chapman, and William W. Smith, A basis for the ring of polynomials integer-valued on prime numbers, in: Daniel Anderson (ed.), Factorization in integral domains, Lecture Notes in Pure and Appl. Math. 189, Dekker, New York, 1997.

Gheorghe Coserea, Table of n, a(n) for n = 1..541
F. Bencherif, Sur une propriété des polynômes de Stirling, 26th Journées Arithmétiques, July 6-10, 2009, Université Jean Monnet, Saint-Etienne, France. [From _Jonathan Sondow_, Jul 23 2009]
M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (2000), 783-799.
Paul-Jean Cahen, and J. L. Chabert, What You Should Know About Integer-Valued Polynomials, The American Mathematical Monthly, 123 (No. 4, 2016), 311-337.
J.-L. Chabert, Integer-valued polynomials on prime numbers and logarithm power expansion, European J. Combinatorics 28 (2007) 754-761. [From _Jonathan Sondow_, Jul 23 2009]
J. L. Chabert, About polynomials whose divided differences are integer-valued on prime numbers, ICM 2012 Proceedings, vol. I, pp. 1-7. Complete proceedings. (warning: file size is 26MB). - From _N. J. A. Sloane_, Nov 28 2012
J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences.
Robert M. Guralnick and Martin Lorenz, Orders of Finite Groups of Matrices, in: William Chin, James Osterburg and Declan Quinn (eds.), Groups, Rings and Algebras, Contemporary Mathematics, Vol. 420 (2006), pp. 141-161; arXiv preprint, arXiv:math/0511191 [math.GR], 2005; author's link.
K. Johnson, The action of the stable operations of complex K-theory on coefficient groups, Illinois J. Math. 28(1), 1984, pp. 57-63. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
K. Johnson, The invariant subalgebra and anti-invariant submodule of K_*K_{(p)}, Jour. of K-theory 2(1), 2008, 123-145. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
Hermann Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196-202. ( = Ges. Abh., pp. 212-218, Chelsea, New York, 1967.)
Issai Schur, Über eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1905), 77-91. ( = Ges. Abh., Bd. 1, pp. 128-142, Springer-Verlag, Berlin-Heidelberg-New York, 1973.)
J.-P. Serre, Bounds for the orders of the finite subgroups of G(k), Group Representation Theory (eds. M. Geck, D. Testerman, J. Thevenaz), EPFL Press, Lausanne, 2006, 405-450.
Wikipedia, Bhargava factorial.
Gregory Gerard Wojnar, Daniel Sz. Wojnar, and Leon Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017. See p. 14.

Cf. A002207, A075264, A075266, A075267.
a(n) = n!*A163176(n). - Jonathan Sondow, Jul 23 2009
Cf. A202318.
Appears in A163972. - Johannes W. Meijer, Oct 16 2009
Cf. A001898, A007814, A163940, A186430, A202339, A212429, A346046.

A053657 := proc(n) local P,p,q,s,r;
P := select(isprime,[$2..n]); r:=1;
for p in P do s := 0; q := p-1;
do if q > (n-1) then break fi;
s := s + iquo(n-1,q); q := q*p; od;
r := r * p^s; od; r end: # Peter Luschny, Jul 26 2009
ser := series((y/(exp(y)-1))^x, y, 20): a := n -> denom(coeff(ser, y, n-1)):
seq(a(n), n=1..19); # Peter Luschny, May 13 2019

m = 16; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, m}]]];
a[n_, k_] := Denominator[ Coefficient[s, x^n*z^k]];
Prepend[Apply[LCM, Table[a[n,k], {n,m}, {k,n}], {1}], 1]
(* Jean-François Alcover, May 31 2011 *)
a[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[ Range[n] ]}]; Array[a, 30] (* Jean-François Alcover, Nov 22 2016 *)

{a(n)=local(X=x+x^2*O(x^n),D);D=1;for(j=0,n-1,D=lcm(D,denominator( polcoeff(polcoeff((-log(1-X)/x)^z+z*O(z^j),j,z),n-1,x))));return(D)} /* Paul D. Hanna, Jun 27 2005 */

{a(n)=prod(i=1,#factor(n!)~,prime(i)^sum(k=0,#binary(n), floor((n-1)/((prime(i)-1)*prime(i)^k))))} /* Paul D. Hanna, Jun 27 2005 */

S(n, p) = {
     my(acc = 0, tmp = p-1);
     while (tmp < n, acc += floor((n-1)/tmp); tmp *= p);
     return(acc);
};
a(n) = {
     my(rv = 1);
     forprime(p = 2, n, rv *= p^S(n,p));
     return(rv);
};
vector(17, i, a(i))  \\ Gheorghe Coserea, Aug 24 2015

a(2n) = 2*a(2n-1). - Jonathan Sondow, Jul 23 2009
a(2*n+1) = 24^n * Product_{i=1..n} A202318(i). - Vladimir Shevelev, Dec 17 2011
For n>=0, A007814(a(n+1)) = n+A007814(n!). - Vladimir Shevelev, Dec 28 2011
a(n) = denominator([y^(n-1)] (y/(exp(y)-1))^x). - Peter Luschny, May 13 2019
Sum_{n>=1} 1/a(n) = A346046. - Amiram Eldar, Jul 02 2023

More terms from Paul D. Hanna, Jun 27 2005

A213449 Denominators of higher order Bernoulli numbers.

Original entry on oeis.org

1, 12, 240, 4032, 34560, 101376, 50319360, 6635520, 451215360, 42361159680, 1471492915200, 1758147379200, 417368899584000, 15410543984640, 141874849382400, 28026642660065280, 922166952040857600, 19725496300339200, 2163255728265599385600, 36926129074234982400

Offset: 0

N. J. A. Sloane, Jun 12 2012

See Nørlund for precise definition.

The 'higher order Bernoulli numbers' considered here are the values of the 'higher order Bernoulli polynomials' evaluated at x=1 (and not at x=0, which would make things boring as x is a factor of these polynomials for n>0). This can be seen as an argument that the definition of the classical Bernoulli numbers as the values of the classical Bernoulli polynomials at x=1 better fits into the general picture than the often used definition as the values at x=0. - Peter Luschny, Oct 01 2016

			From _Peter Luschny_, Oct 01 2016: (Start)
The sequence of polynomials starts:
1,
(1/12*(3*x-1))*x,
(1/240*(15*x^3-30*x^2+5*x+2))*x,
(1/4032*(63*x^5-315*x^4+315*x^3+91*x^2-42*x-16))*x,
(1/34560*(135*x^7-1260*x^6+3150*x^5-840*x^4-2345*x^3-540*x^2+404*x+144))*x. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer, 1924, p. 459.

Cf. A000367 (numerators of the polynomials evaluated at x=1 at even indices).

Bisection (even indices) of A001898.

B := proc(v,n) option remember; `if`(v = 0,1,
simplify(-(n/v)*add((-1)^s*binomial(v,s)*bernoulli(s)*B(v-s,n),s=1..v))) end:
A213449 := n -> denom(B(2*n, k)):
seq(A213449(n), n=0..19); # Peter Luschny, Oct 01 2016

Table[NorlundB[2n, x] // Together // Denominator, {n, 0, 19}] (* Jean-François Alcover, Jun 29 2019 *)

Name corrected and more terms added by Peter Luschny, Oct 01 2016

A284849 Number of zeros strictly inside the unit circle of the Bernoulli polynomial B(n,x).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 5, 6, 6, 5, 5, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3

Offset: 0

Michel Lagneau, May 09 2017

nonn

The n-th Bernoulli polynomial is defined by the exponential generating function:
t*exp(x*t)/(exp(t)-1) = Sum_{n>=0} bernoulli(n,x)/n!*t^n.
The first few Bernoulli polynomials are:
B(0,x) = 1
B(1,x) = x - 1/2
B(2,x) = x^2 - x + 1/6
B(3,x) = x^3 - 3/2 x^2 + 1/2 x
B(4,x)= x^4 - 2x^3 + x^2 - 1/30
Conjecture: for n > 63, a(n) = 3 for n odd and a(n) = 4 otherwise. - Charles R Greathouse IV, May 09 2017

			a(6) = 4 because the zeros of B(6,x) = x^6 - 3x^5 + 5/2 x^4 - 1/2 x^2 + 1/42 are:
x1 = -0.2728865...-0.06497293...*i,
x2 = -0.2728865...+0.06497293...*i,
x3 =  0.2475407...,
x4 =  0.7524592...,
x5 =  1.272886...-.06497293...*i,
x6 =  1.272886...+.06497293...*i
with four roots x1, x2, x3 and x4 in the unit circle.

Eric Weisstein's World of Mathematics, Bernoulli Polynomial

Cf. A001898, A027641, A027642.

for n from 0 to 90 do:it:=0:
y:=[fsolve(expand(bernoulli(n,x)),x,complex)]:for m from 1 to nops(y) do:if abs(y[m])<1 then it:=it+1:else fi:od: printf(`%d, `, it):od:

a(n)=my(v=polroots(bernpol(n))); sum(i=1,#v,abs(v[i])<1) \\ Charles R Greathouse IV, May 09 2017

Conjectures from Colin Barker, Jan 22 2020: (Start)
G.f.: x*(1 + 2*x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 - x^12 - x^13 + x^14 - x^15 - x^16 - x^17 - x^18 - x^20 + x^22 - x^24 + x^26 - x^28 + x^30 - x^32) / ((1 - x)*(1 + x)).
a(n) = a(n-2) for n>33.
(End)

Showing 1-7 of 7 results.

cp's OEIS Frontend

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).

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A100616 Let B(n)(x) be the Bernoulli polynomials as defined in A001898, with B(n)(1) equal to the usual Bernoulli numbers A027641/A027642. Sequence gives denominators of B(n)(2).

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A100655 Triangle read by rows giving coefficients in Bernoulli polynomials as defined in A001898, after multiplication by the common denominators A001898(n).

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A137524 Triangular sequence from coefficients of the umbral calculus expansion of a Golden -Mean Bernoulli function(A001898): p(x,t)=t*phi^(x*t)/(phi^t - 1), where the golden ratio replaces "e".

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A053657 a(n) = Product_{p prime} p^{ Sum_{k>=0} floor[(n-1)/((p-1)p^k)]}.

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A213449 Denominators of higher order Bernoulli numbers.

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A137524 Triangular sequence from coefficients of the umbral calculus expansion of a Golden -Mean Bernoulli function(A001898): p(x,t)=tphi^(xt)/(phi^t - 1), where the golden ratio replaces "e".