A100655 -id:A100655 - cp's OEIS frontend

A001898 Denominators of Bernoulli polynomials B(n)(x).

1, 2, 12, 8, 240, 96, 4032, 1152, 34560, 7680, 101376, 18432, 50319360, 7741440, 6635520, 884736, 451215360, 53084160, 42361159680, 4459069440, 1471492915200, 140142182400, 1758147379200, 152882380800, 417368899584000, 33389511966720, 15410543984640

Offset: 0

N. J. A. Sloane

nonn

			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.

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. Nørlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 459.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
Index entries for sequences related to Bernoulli numbers.

Cf. A027641, A027642, A100615, A100616, A100655.

B:=bernoulli; b:=proc(s) option remember; local t; global r; if s=0 then RETURN(1); fi; expand((-r/s)*add( (-1)^t*binomial(s,t)*B(t)*b(s-t),t=1..s)); end; [seq(denom(b(n)),n=0..30)];

B[s_] := B[s] = If[s == 0, 1, (-x/s)*Sum[(-1)^t*Binomial[s, t]*
   BernoulliB[t]*B[s - t], {t, 1, s}]] // Factor;
a[n_] := If[n == 0, 1, B[n] // First // Denominator];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 24 2022 *)

These Bernoulli polynomials B(s) = B(s)(x) are defined by: B(0) = 1; B(s) = (-x/s)*Sum_{t=1..s} (-1)^t*binomial(s, t)*Bernoulli(t)*B(s-t), where Bernoulli(t) are the usual Bernoulli numbers A027641/A027642. Also B(s)(1) = Bernoulli(s).

Entry revised Dec 03 2004

A290030 Leading coefficients of numerators of Norlund's B_{nu}^(n) polynomials (Nørlund, Tafel 5, p. 459).

Original entry on oeis.org

1, -1, 3, -1, 15, -3, 63, -9, 135, -15, 99, -9, 12285, -945, 405, -27, 6885, -405, 161595, -8505, 1403325, -66825, 419175, -18225, 24877125, -995085, 229635, -8505, 528525, -18225, 26101845, -841995, 214708725, -6506325, 1148175, -32805, 31479513975, -850797675

Offset: 0

Gregory Gerard Wojnar, Jul 17 2017

sign

Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See arXiv:1706.08381 [math,GM], 2017.] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N)= ((-1)^D/(D-1)!)*(D-N)*N^chi(D)*g_D(N) where chi(D) := (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). The coefficients of the g_D(N) are polynomials in D of the form k_n(D)=(1/Q(n))*(D+t(n))^delta(n)*D^chi(n+1)*u_n(D) where Q(n)=A053657(n), t(n):=2 ceiling(n/2)+1, delta(n):= (1 if n is odd, 2 if n is even). The leading coefficients of u_n(D) are a(n).

Gregory Gerard Wojnar, Table of n, a(n) for n = 0..187
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 459.
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

Cf. A053657, A100655.

a[n_] := NorlundB[n, x] // Together // Numerator // Coefficient[#, x, n]&;
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Jun 30 2019 *)

[A100655_row(n)[n] for n in (0..37)] # Peter Luschny, Jul 01 2019

A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 96, 192, 1152, 768, 1536, 3072, 18432, 36864, 221184, 147456, 884736, 1769472, 10616832, 21233664, 637009920, 424673280, 2548039680, 5096079360, 152882380800, 61152952320, 366917713920, 81537269760, 163074539520, 326149079040, 1956894474240

Offset: 0

Peter Luschny, Feb 06 2021

nonn

The challenge is to characterize the sequence purely arithmetically, i.e., without reference to the Eulerian numbers or the Bernoulli polynomials.

András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences In: J. W. Brewer, S. Glaz, W. J. Heinzer, B. M. Olberding (eds), Multiplicative Ideal Theory in Commutative Algebra. Springer, Boston, MA., 2006.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, arXiv:1705.03857 [math.NT] 2017, Amer. Math. Monthly.

Cf. A100655, A053657 (Minkowski), A341107, A341108, A318256, A144845, A163176, A201637 (Eulerian2), A036689, A324370, A007947, A324369, A195441.

Epoly := proc(n, x) add(combinat:-eulerian2(n, k)*binomial(x+k, 2*n), k = 0..n) / mul(j-x, j = 1..n): simplify(expand(%)) end:
seq(denom(Epoly(n, x)) / (n!*denom(bernoulli(n, x))), n = 0..30);

A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k,0,n}],{p, Prime[Range[n]]}];
A144845[n_] := Denominator[Together[BernoulliB[n, x]]];
A163176[n_] := A053657[n] / n!;
Table[(n + 1) A163176[n + 1] / A144845[n], {n, 0, 30}]

def A341109(n): # uses[A341108, A318256]
    return A341108(n)//A318256(n)
print([A341109(n) for n in (0..30)])

a(n) = A053657(n+1)/(n!*A144845(n)).
a(n) = (n+1)*A163176(n+1)/A144845(n).
a(n) = A341108(n)/A318256(n).
a(n) = A341107(n)*A324369(n+1).
a(n) = A341108(n)/A324370(n+1).
a(n) = A341108(n)*A007947(n+1)/A144845(n).
a(n) = A341108(n)*A324369(n+1)/A195441(n).
prime(n) divides a(k) for k >= A036689(n).
2^(n-1) divides exactly a(n) for n >= 2.

cp's OEIS Frontend

A001898 Denominators of Bernoulli polynomials B(n)(x).

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A290030 Leading coefficients of numerators of Norlund's B_{nu}^(n) polynomials (Nørlund, Tafel 5, p. 459).

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A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

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cp's OEIS Frontend

A001898 Denominators of Bernoulli polynomials B(n)(x).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A290030 Leading coefficients of numerators of Norlund's B_{nu}^(n) polynomials (Nørlund, Tafel 5, p. 459).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

A341109 a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.