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.

Showing 1-5 of 5 results.

A157779 Numerator of Bernoulli(n, 1/2).

Original entry on oeis.org

1, 0, -1, 0, 7, 0, -31, 0, 127, 0, -2555, 0, 1414477, 0, -57337, 0, 118518239, 0, -5749691557, 0, 91546277357, 0, -1792042792463, 0, 1982765468311237, 0, -286994504449393, 0, 3187598676787461083, 0, -4625594554880206790555, 0, 16555640865486520478399, 0
Offset: 0

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Author

N. J. A. Sloane, Nov 08 2009

Keywords

Comments

Included for completeness, normally alternating zeros like this are omitted. A001896 is the official version of this sequence.
The sequence {a(n)/A141459(n)} gives the generalized Bernoulli numbers B[2,1] obtained from the generalized Stirling2 triangle S3[2,1] = A154537. See the formula section. - Wolfdieter Lang, Apr 27 2017

Links

Crossrefs

For denominators see A157780 and A141459.

Programs

  • Mathematica
    Numerator[BernoulliB[Range[0,40],1/2]] (* Harvey P. Dale, May 04 2013 *)
  • PARI
    a(n) = numerator(subst(bernpol(n, x), x, 1/2)); \\ Altug Alkan, Jul 05 2016
  • Sage
    def A157779_list(size):
    f = x / sum(x^(n*2+1)/factorial(n*2+1) for n in (0..2*size))
    t = taylor(f, x, 0, size)
    return [(factorial(n)*s).numerator() for n,s in enumerate(t.list())]
print(A157779_list(33)) # Peter Luschny, Jul 05 2016

Formula

Let P(x) = Sum_{n>=0} x^(2*n+1)/(2*n+1)!; then a(n) = numerator( n! [x^n] x/P(x) ). - Peter Luschny, Jul 05 2016
a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A154537(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A145901(n, k). The denominators are in A141459. r(n) = B[2,1](n) = 2^n*B(n, 1/2) with the Bernoulli polynomials A196838/A196839 or A053382/A053383. - Wolfdieter Lang, Apr 27 2017
a(n) = numerator(-(1-2^(1-n))*Bernoulli(n)). - Fabián Pereyra, Dec 31 2022

A224783 Denominator of Bernoulli(n,1/2) - Bernoulli(n,0).

Original entry on oeis.org

1, 2, 4, 1, 16, 1, 64, 1, 256, 1, 1024, 1, 4096, 1, 16384, 1, 65536, 1, 262144, 1, 1048576, 1, 4194304, 1, 16777216, 1, 67108864, 1, 268435456, 1, 1073741824, 1, 4294967296, 1, 17179869184, 1, 68719476736, 1, 274877906944, 1, 1099511627776
Offset: 0

Views

Author

Paul Curtz, Apr 17 2013

Keywords

Comments

See A157779 and A157780 for values of Bernoulli(n,1/2), and A027641 and A027642 for values of Bernoulli(n,0).
B(n,1/2) - B(n,0) = 0, 1/2, -1/4, 0, 1/16, 0, -3/64, 0, 17/256, 0, -155/1024, 0, 2073/4096, 0, -38227/16384,... for n>=0.
The sequence of numerators is 0, 1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227, 0, 929569, 0, -28820619, 0, 1109652905,...and appears to contain a mix of A001469 and A036968.

Examples

			a(0) = 1-1, a(1) = 0+1/2, a(2) = -1/12-1/6=-1/4.

Links

Crossrefs

Cf. A001469, A027641, A027642, A036968, A059222, A157779, A157780.

Programs

  • Maple
    A224783 := proc(n)
    bernoulli(n,1/2)-bernoulli(n) ;
    denom(%) ;
end proc: # R. J. Mathar, Apr 25 2013
  • Mathematica
    Table[Denominator[BernoulliB[n, 1/2] - BernoulliB[n, 0]], {n, 0, 50}] (* Vincenzo Librandi, Mar 19 2014 *)
  • PARI
    Vec((4*x^5-9*x^3-x^2+2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)) + O(x^100)) \\ Colin Barker, Mar 20 2014

Formula

a(n) = A059222(n+1) if n <> 1.
From Colin Barker, Mar 19 2014: (Start)
G.f.: (4*x^5-9*x^3-x^2+2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)).
a(n) = 5*a(n-2)-4*a(n-4) for n>5.
a(n) = (1+(-2)^n-(-1)^n+2^n)/2 for n>1. (End).

A335947 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 7, 0, -1, 0, 1, 0, 7, 0, -5, 0, 1, -31, 0, 7, 0, -5, 0, 1, 0, -31, 0, 49, 0, -7, 0, 1, 127, 0, -31, 0, 49, 0, -7, 0, 1, 0, 381, 0, -31, 0, 147, 0, -3, 0, 1, -2555, 0, 381, 0, -155, 0, 49, 0, -15, 0, 1
Offset: 0

Views

Author

Peter Luschny, Jul 01 2020

Keywords

Comments

The polynomials form an Appell sequence.
The parity of n equals the parity of b(n, x). The Bernoulli polynomials do not possess this property.

Examples

			First few polynomials are:
b_0(x) = 1;
b_1(x) = x;
b_2(x) = -(1/12) + x^2;
b_3(x) = -(1/4)*x + x^3;
b_4(x) = (7/240) - (1/2)*x^2 + x^4;
b_5(x) = (7/48)*x - (5/6)*x^3 + x^5;
b_6(x) = -(31/1344) + (7/16)*x^2 - (5/4)*x^4 + x^6;
Normalized by A335949:
b_0(x) = 1;
b_1(x) = x;
b_2(x) = (-1 + 12*x^2) / 12;
b_3(x) = (-x + 4*x^3) / 4;
b_4(x) = (7 - 120*x^2 + 240*x^4) / 240;
b_5(x) = (7*x - 40*x^3 + 48*x^5) / 48;
b_6(x) = (-31 + 588*x^2 - 1680*x^4 + 1344*x^6) / 1344;
b_7(x) = (-31*x + 196*x^3 - 336*x^5 + 192*x^7) / 192;
Triangle starts:
[0] 1;
[1] 0,   1;
[2] -1,  0,   1;
[3] 0,   -1,  0,   1;
[4] 7,   0,   -1,  0,   1;
[5] 0,   7,   0,   -5,  0,  1;
[6] -31, 0,   7,   0,   -5, 0,   1;
[7] 0,   -31, 0,   49,  0,  -7,  0,  1;
[8] 127, 0,   -31, 0,   49, 0,   -7, 0,  1;
[9] 0,   381, 0,   -31, 0,  147, 0,  -3, 0, 1;

Crossrefs

Cf. A335948 (denominators), A335949 (denominators of the polynomials).
Cf. A157779 (column 0), A001896 (column 0 at even indices only).
Cf. A164555/A027642, A157781/A157782, A157779/A157780, A196838/A196839.

Programs

  • Maple
    b := (n,x) -> bernoulli(n, x+1/2):
A335947row := n -> seq(numer(coeff(b(n,x), x, k)), k = 0..n):
seq(A335947row(n), n = 0..10);

Formula

b(n, 1/2) = Bernoulli(n, 1) = A164555(n)/A027642(n).
b(n, -1) = Bernoulli(n, -1/2) = A157781(n)/A157782(n).
b(n, 0) = Bernoulli(n, 1/2) = A157779(n)/A157780(n).
b(n, x) = Bernoulli(n, x + 1/2).

A335948 T(n, k) = denominator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 240, 1, 2, 1, 1, 1, 48, 1, 6, 1, 1, 1344, 1, 16, 1, 4, 1, 1, 1, 192, 1, 48, 1, 4, 1, 1, 3840, 1, 48, 1, 24, 1, 3, 1, 1, 1, 1280, 1, 16, 1, 40, 1, 1, 1, 1, 33792, 1, 256, 1, 32, 1, 8, 1, 4, 1, 1, 1, 3072, 1, 256, 1, 32, 1, 8, 1, 12, 1, 1
Offset: 0

Views

Author

Peter Luschny, Jul 01 2020

Keywords

Comments

See A335947 for formulas and references concerning the polynomials.

Examples

			First few polynomials are:
b_0(x) = 1;
b_1(x) = x;
b_2(x) = -(1/12) + x^2;
b_3(x) = -(1/4)*x + x^3;
b_4(x) = (7/240) - (1/2)*x^2 + x^4;
b_5(x) = (7/48)*x - (5/6)*x^3 + x^5;
b_6(x) = -(31/1344) + (7/16)*x^2 - (5/4)*x^4 + x^6;
Triangle starts:
1;
1,     1;
12,    1,    1;
1,     4,    1,   1;
240,   1,    2,   1,  1;
1,     48,   1,   6,  1,  1;
1344,  1,    16,  1,  4,  1,  1;
1,     192,  1,   48, 1,  4,  1, 1;
3840,  1,    48,  1,  24, 1,  3, 1, 1;
1,     1280, 1,   16, 1,  40, 1, 1, 1, 1;
33792, 1,    256, 1,  32, 1,  8, 1, 4, 1, 1;

Crossrefs

Cf. A335947 (numerators), A157780 (column 0), A033469 (column 0 even indices only).

A326582 A signed variant of A309132.

Original entry on oeis.org

1, 1, 1, 4, 1, 9, 1, 16, 27, 25, 1, 36, 1, 49, -75, 64, 1, 81, -1, 100, 49, 121, -1, 144, 125, 169, -243, 196, 1, 225, -1, 256, 363, 289, -1225, 324, 1, 361, -169, 400, 1, 441, -1, 484, 135, 529, -1, 576, 343, 625, -867, 676, 1, 729, -3025, 784, 361, 841, -1
Offset: 0

Views

Author

Peter Luschny, Jul 15 2019

Keywords

Comments

See A309132 for background and conjectures.

Crossrefs

Cf. A309132, A157779, A157780.

Programs

  • Maple
    nB := n -> numer(bernoulli(n-1,1/2)): dB := n -> denom(bernoulli(n-1,1/2)):
R := n -> n/(nB(n) + dB(n)/n): a := n -> numer(R(n+1)/4^irem(n,2)):
seq(a(n), n=0..58);

Formula

a(n) = numerator(R(n+1)/4^mod(n,2)) with R(n) = n/(nB(n) + dB(n)/n) and nB(n) = numerator(B(n-1, 1/2)), dB(n) = denominator(B(n-1, 1/2)) where B(n, x) denotes the Bernoulli polynomials.
|a(2*n)| = A309132(2*n + 1) for n >= 0.
a(2*n+1) = (n + 1)^2 for n >= 0.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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