cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A366152 a(n) = denominator(denominator(Bernoulli'(n, x)) / denominator(Bernoulli(n, 1))).

Original entry on oeis.org

1, 2, 6, 1, 30, 1, 42, 1, 10, 1, 66, 1, 2730, 1, 2, 1, 170, 1, 798, 1, 110, 1, 46, 1, 546, 1, 2, 1, 870, 1, 14322, 1, 170, 1, 2, 1, 1919190, 1, 2, 1, 4510, 1, 1806, 1, 46, 1, 94, 1, 1326, 1, 22, 1, 530, 1, 798, 1, 290, 1, 118, 1, 56786730, 1, 2, 1, 34, 1, 64722
Offset: 0

Views

Author

Peter Luschny, Oct 13 2023

Keywords

Crossrefs

Cf. A324370/A027642, A366570 (numerator), A366426/A366427 (2nd derivative).

Programs

  • Maple
    seq(denom(denom(diff(bernoulli(n, x), x))/denom(bernoulli(n, 1))), n = 0..66);
  • PARI
    a(n) = denominator(lcm(apply(denominator, Vec(deriv(bernpol(n)))))/denominator(subst(bernpol(n, x), x, 1))); \\ Michel Marcus, Oct 14 2023

Formula

a(n) = denominator(A324370(n) / A027642(n)).

A366570 a(n) = numerator(denominator(Bernoulli'(n, x)) / denominator(Bernoulli(n, 1))).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 6, 1, 10, 1, 6, 1, 210, 5, 2, 1, 30, 5, 210, 7, 110, 5, 30, 1, 546, 7, 14, 1, 30, 1, 462, 77, 1190, 35, 6, 1, 51870, 455, 70, 7, 2310, 55, 2310, 7, 322, 35, 210, 1, 6630, 221, 286, 11, 330, 55, 798, 19, 290, 5, 30, 1, 930930, 5005, 1430
Offset: 0

Views

Author

Peter Luschny, Oct 13 2023

Keywords

Crossrefs

Cf. A324370/A027642, A366152 (denominator), A366426/A366427 (2nd derivative).

Programs

  • Maple
    seq(numer(denom(diff(bernoulli(n, x), x))/denom(bernoulli(n, 1))), n = 0..66);
  • PARI
    a(n) = numerator(lcm(apply(denominator, Vec(deriv(bernpol(n)))))/denominator(subst(bernpol(n, x), x, 1))); \\ Michel Marcus, Oct 14 2023

Formula

a(n) = numerator(A324370(n) / A027642(n)).

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

Views

Author

Peter Luschny, Feb 06 2021

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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}]
  • Sage
    def A341109(n): # uses[A341108, A318256]
    return A341108(n)//A318256(n)
print([A341109(n) for n in (0..30)])

Formula

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.

A366572 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli''(n, x)).

Original entry on oeis.org

1, 2, 6, 2, 30, 6, 42, 6, 10, 10, 66, 6, 2730, 210, 2, 6, 510, 10, 798, 42, 110, 330, 138, 2, 546, 546, 2, 2, 870, 30, 14322, 462, 170, 510, 6, 2, 1919190, 51870, 2, 6, 13530, 110, 1806, 42, 46, 690, 1410, 2, 1326, 1326, 22, 66, 1590, 10, 798, 798, 290, 870
Offset: 0

Views

Author

Peter Luschny, Oct 13 2023

Keywords

Crossrefs

Cf. A144845/A366168, A366571, A144845/A324370 (1st derivative).

Programs

  • Maple
    seq(denom(bernoulli(n, x))/denom(diff(diff(bernoulli(n, x), x),x)), n=0..100);
  • PARI
    a(n) = lcm(apply(denominator, Vec(bernpol(n))))/lcm(apply(denominator, Vec(deriv(deriv(bernpol(n)))))); \\ Michel Marcus, Oct 14 2023

Formula

a(n) = A144845(n) / A366168(n).

A366571 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli'(n, x)).

Original entry on oeis.org

1, 2, 6, 1, 30, 1, 42, 1, 10, 1, 66, 1, 2730, 1, 2, 3, 170, 1, 798, 1, 110, 3, 46, 1, 546, 1, 2, 1, 870, 1, 14322, 1, 170, 3, 2, 1, 1919190, 1, 2, 3, 4510, 1, 1806, 1, 46, 15, 94, 1, 1326, 1, 22, 3, 530, 1, 798, 1, 290, 3, 118, 1, 56786730, 1, 2, 3, 34, 5, 64722
Offset: 0

Views

Author

Peter Luschny, Oct 13 2023

Keywords

Crossrefs

Cf. A144845/A324370, A366572, A144845/A366168 (2nd derivative).

Programs

  • Maple
    seq(denom(bernoulli(n, x))/denom(diff(bernoulli(n, x), x)), n = 0..66);
  • PARI
    a(n) = lcm(apply(denominator, Vec(bernpol(n))))/lcm(apply(denominator, Vec(deriv(bernpol(n))))); \\ Michel Marcus, Oct 14 2023

Formula

a(n) = A144845(n) / A324370(n).

A366573 a(n) = denominator(Bernoulli'(n, x)) / denominator(Bernoulli''(n, x)).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 6, 1, 10, 1, 6, 1, 210, 1, 2, 3, 10, 1, 42, 1, 110, 3, 2, 1, 546, 1, 2, 1, 30, 1, 462, 1, 170, 3, 2, 1, 51870, 1, 2, 3, 110, 1, 42, 1, 46, 15, 2, 1, 1326, 1, 22, 3, 10, 1, 798, 1, 290, 3, 2, 1, 930930, 1, 2, 3, 34, 5, 966, 1, 2, 3, 110, 1
Offset: 0

Views

Author

Peter Luschny, Oct 13 2023

Keywords

Crossrefs

Cf. A324370/A366168, A366572, A144845/A366168.

Programs

  • Maple
    seq(denom(diff(bernoulli(n, x), x))/denom(diff(diff(bernoulli(n, x), x),x)), n = 0..100);
  • PARI
    a(n) = lcm(apply(denominator, Vec(deriv(bernpol(n)))))/ lcm(apply(denominator, Vec(deriv(deriv(bernpol(n)))))); \\ Michel Marcus, Oct 14 2023

Formula

a(n) = A324370(n) / A366168(n).
Previous Showing 21-26 of 26 results.

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