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 1-4 of 4 results.

A156201 Numerator of Euler(n, 1/8).

Original entry on oeis.org

1, -3, -7, 117, 497, -15123, -95767, 4116837, 34741217, -1921996323, -20273087527, 1370953667157, 17352768515537, -1386843017916723, -20479521468959287, 1888542637550347077, 31872138933891307457, -3331009898404800736323, -63243057486503656319047
Offset: 0

Views

Author

N. J. A. Sloane, Nov 07 2009

Keywords

Links

Crossrefs

For denominators see A001018. Cf. A000813.

Programs

  • Maple
    p := proc(n) local j; 2*I*(1+add(binomial(n,j)*polylog(-j,I)*4^j, j=0..n)) end:  A156201 := n -> Im(p(n));
seq(A156201(i), i=0..10);  # Peter Luschny, Apr 29 2013
  • Mathematica
    Table[EulerE[n, 1/8] // Numerator, {n, 0, 18}] (* Jean-François Alcover, Apr 30 2013 *)

Formula

a(n) = Im(2*i*(1+Sum_{j=0..n} (binomial(n,j)*Li_{-j}(i)*4^j))). - Peter Luschny, Apr 29 2013
G.f.: conjecture T(0)/(1+3*x), where T(k) = 1 - 16*x^2*(k+1)^2/(16*x^2*(k+1)^2 + (1+3*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013
a(n) = (-4)^n*skp(n, 3/4), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014
a(n) = 2^(4*n+1)*(zeta(-n,1/16)-zeta(-n, 9/16)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
From Emanuele Munarini, Aug 22 2022: (Start)
E.g.f.: (2*e^t)/(e^(8*t)+1).
E.g.f. for the sequence of the absolute values: (cos(3*t)+sin(3*t))/cos(4*t).
|a(2*n)| = Sum_{k=0..n} binomial(2*n,2*k) (-1)^k 4^(2*n-2*k) 3^(2*k) |E(2*n-2k)|
|a(2*n+1)| = Sum_{k=0..n} binomial(2*n+1,2*k+1) (-1)^k 4^(2*n-2*k) 3^(2*k+1) |E(2*n-2*k)|
where the E(n)'s are the Euler numbers (A122045).
(End)

A378066 Array read by ascending antidiagonals: A(n, k) = (-2*n)^k * Euler(k, (n - 1)/(2*n)) for n >= 1 and A(0, k) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -3, -2, 1, 1, 1, -8, -11, 0, 1, 1, 1, -15, -26, 57, 16, 1, 1, 1, -24, -47, 352, 361, 0, 1, 1, 1, -35, -74, 1185, 1936, -2763, -272, 1, 1, 1, -48, -107, 2976, 6241, -38528, -24611, 0, 1
Offset: 0

Views

Author

Peter Luschny, Nov 15 2024

Keywords

Comments

This is the counterpart of A377666, where A(1, n) are the secant numbers A122045(n). Here A(1, n) are the tangent numbers A155585(n).

Examples

			Array starts:
  [0]  1, 1,   1,    1,     1,     1,        1, ...  A000012
  [1]  1, 1,   0,   -2,     0,    16,        0, ...  A155585
  [2]  1, 1,  -3,  -11,    57,   361,    -2763, ...  A188458
  [3]  1, 1,  -8,  -26,   352,  1936,   -38528, ...  A000810
  [4]  1, 1, -15,  -47,  1185,  6241,  -230895, ...  A000813
  [5]  1, 1, -24,  -74,  2976, 15376,  -906624, ...  A378065
  [6]  1, 1, -35, -107,  6265, 32041, -2749355, ...
  [7]  1, 1, -48, -146, 11712, 59536, -6997248, ...

Crossrefs

Rows: A000012, A155585, A188458, A000810, A000813, A378065.
Columns: A005563 (k=2), A080663 (k=3), A378064 (k=4).
Cf. A378063 (main diagonal), A377666 (secant), A081658 (column generating polynomials).

Programs

  • Maple
    A := (n, k) -> ifelse(n = 0, 1, (-2*n)^k * euler(k, (n - 1) / (2*n))):
for n from 0 to 7 do seq(A(n, k), k = 0..9) od; # row by row
# Alternative:
A := proc(n, k) local j; add(binomial(k, j)*euler(j, 1/2)*(-2*n)^j, j = 0..k) end: seq(seq(A(n - k, k), k = 0..n), n = 0..10);
# Using generating functions:
egf := n -> exp(x)/cosh(n*x): ser := n -> series(egf(n), x, 14):
row := n -> local k; seq(k!*coeff(ser(n), x, k), k = 0..7):
seq(lprint(row(n)), n = 0..7);

Formula

A(n, k) = k! * [x^k] exp(x)/cosh(n*x).
A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 1/2) *(-2*n)^j.

A001728 Expansion of cos x / cos 4x.

Original entry on oeis.org

1, 15, 1185, 230895, 83860545, 48942778575, 41893214676705, 49441928730798255, 76946148390480577665, 152682246738275154625935, 376229085883258481118811425, 1127131459348047116845576437615, 4034538627668651235506091915425985
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Bisection of A000813.

Programs

  • Mathematica
    nn = 22; t = Range[0, nn]! CoefficientList[Series[Cos[x]/Cos[4 x], {x, 0, nn}], x]; Take[t, {1, nn, 2}];

A156205 Numerator of Euler(n, 3/8).

Original entry on oeis.org

1, -1, -15, 47, 1185, -6241, -230895, 1704527, 83860545, -796079041, -48942778575, 567864586607, 41893214676705, -574448847467041, -49441928730798255, 782259922208550287, 76946148390480577665, -1379749466246228538241, -152682246738275154625935
Offset: 0

Views

Author

N. J. A. Sloane, Nov 07 2009

Keywords

Links

Crossrefs

For denominators see A001018. Cf. A000813.

Programs

  • Maple
    p := proc(n) local j; 2*I*(1+add(binomial(n,j)*polylog(-j,I)*4^j, j=0..n)) end:  A156205 := n -> (-1)^(n+1)*Re(p(n));
seq(A156205(i),i=0..11);  # Peter Luschny, Apr 29 2013
  • Mathematica
    Numerator[EulerE[Range[0,20],3/8]] (* Vincenzo Librandi, May 04 2012 *)

Formula

a(n) = (-1)^(n+1)*Re(2*I*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*4^j))). - Peter Luschny, Apr 29 2013
a(n) = (-4)^n*skp(n, 1/4), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014
Showing 1-4 of 4 results.

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

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