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 1-6 of 6 results.

A326481 a(n) = E2_{n}(0) with E2_{n} the polynomials defined in A326480.

Original entry on oeis.org

1, -2, 2, 4, -16, -32, 272, 544, -7936, -15872, 353792, 707584, -22368256, -44736512, 1903757312, 3807514624, -209865342976, -419730685952, 29088885112832, 58177770225664, -4951498053124096, -9902996106248192, 1015423886506852352
Offset: 0

Views

Author

Peter Luschny, Jul 12 2019

Keywords

Comments

For comments see A326480.

Crossrefs

Bisections (up to signs): A000182, A012509.
Cf. A326480, A155585, A326481, A326482, A326483, A326484.
Cf. A063376, A241209.

Programs

  • Maple
    # The function E2(n) is defined in A326480.
seq(subs(x=0, E2(n)), n=0..22);

A326482 a(n) = E2_{n}(-1) with E2_{n} the polynomials defined in A326480.

Original entry on oeis.org

1, -4, 14, -40, 80, -64, -16, -1600, 8960, 29696, -349696, -1423360, 22384640, 89440256, -1903691776, -7615160320, 209865605120, 839460847616, -29088884064256, -116355542548480, 4951498057318400, 19805992204107776, -1015423886490075136
Offset: 0

Views

Author

Peter Luschny, Jul 12 2019

Keywords

Comments

For comments see A326480.

Crossrefs

Cf. A326480, A155585, A326481, A326482, A326483, A326484.

Programs

  • Maple
    # The function E2(n) is defined in A326480.
seq(subs(x=-1, E2(n)), n=0..22);
  • Mathematica
    T[n_, k_] := 2^n n! SeriesCoefficient[4 Exp[x z]/(Exp[z] + 1)^2, {z, 0, n}, {x, 0, k}]; Table[Sum[(-1)^k T[n, k], {k, 0, n}], {n, 0, 22}] (* Jean-François Alcover, Jul 23 2019 *)

A326483 a(n) = 2^n*E2_{n}(1/2) with E2_{n} the polynomials defined in A326480.

Original entry on oeis.org

1, -2, -4, 40, 80, -1952, -3904, 177280, 354560, -25866752, -51733504, 5535262720, 11070525440, -1633165156352, -3266330312704, 635421069967360, 1270842139934720, -315212388819402752, -630424777638805504, 194181169538675507200
Offset: 0

Views

Author

Peter Luschny, Jul 12 2019

Keywords

Comments

For comments see A326480.

Crossrefs

Bisections (up to signs): A002436 (even), A000816 (odd).
Cf. A326480, A155585, A326481, A326482, A326483, A326484.

Programs

  • Maple
    # The function E2(n) is defined in A326480.
seq(subs(x=1/2, 2^n*E2(n)), n=0..22);

Formula

From Emanuele Munarini, Aug 22 2022: (Start)
E.g.f. for the sequence of the absolute values: (1+tan(2*t))/cos(2*t).
|a(2*n)| = 2^(2*n) |E(2*n)|.
|a(2*n+1)| = 2^(2*n+1) Sum_{k=0..n} binomial(2*n+1,2*k) |E(2*k)| T(n-k+1), where the E(n) are the Euler numbers (A122045) and the T(n) are the tangent numbers (A000182). (End)

A326484 a(n) = 2^n*E2_{n}(-1/2) with E2_{n} the polynomials defined in A326480.

Original entry on oeis.org

1, -6, 28, -72, -176, 1824, 11968, -177792, -1062656, 25864704, 155204608, -5535270912, -33211559936, 1633165123584, 9798991003648, -635421070098432, -3812526419542016, 315212388818878464, 1891274332917465088, -194181169538677604352, -1165087017232048848896
Offset: 0

Views

Author

Peter Luschny, Jul 12 2019

Keywords

Comments

For comments see A326480.

Crossrefs

Cf. A326480, A155585, A326481, A326482, A326483, A326484.

Programs

  • Maple
    # The function E2(n) is defined in A326480.
seq(subs(x=-1/2, 2^n*E2(n)), n=0..22);

A326485 T(n, k) = 2^A050605(n) * n! * [x^k] [z^n] (4*exp(x*z))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, -1, 1, 1, -4, 2, 1, 3, -6, 2, -1, 2, 3, -4, 1, -1, -5, 5, 5, -5, 1, 17, -24, -60, 40, 30, -24, 4, 17, 119, -84, -140, 70, 42, -28, 4, -31, 34, 119, -56, -70, 28, 14, -8, 1, -31, -279, 153, 357, -126, -126, 42, 18, -9, 1, 691, -620, -2790, 1020, 1785, -504, -420, 120, 45, -20, 2
Offset: 0

Views

Author

Peter Luschny, Jul 12 2019

Keywords

Comments

These are the coefficients of the generalized Euler polynomials (case m=2) with a different normalization. See A326480 for further comments.

Examples

			Triangle starts:
[0] [  1]
[1] [ -1,    1]
[2] [  1,   -4,   2]
[3] [  1,    3,  -6,    2]
[4] [ -1,    2,   3,   -4,    1]
[5] [ -1,   -5,   5,    5,   -5,    1]
[6] [ 17,  -24, -60,   40,   30,  -24,   4]
[7] [ 17,  119, -84, -140,   70,   42, -28,  4]
[8] [-31,   34, 119,  -56,  -70,   28,  14, -8,  1]
[9] [-31, -279, 153,  357, -126, -126,  42, 18, -9, 1]

Crossrefs

Cf. A326480, A050605.

Programs

  • Maple
    E2n := proc(n) (4*exp(x*z))/(exp(z) + 1)^2;
series(%, z, 48); 2^A050605(n)*n!*coeff(%, z, n) end:
for n from 0 to 9 do PolynomialTools:-CoefficientList(E2n(n), x) od;

A326479 T(n, k) = 2^n * n! * [x^k] [z^n] (exp(z) + 1)^2/(4*exp(x*z)), triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 2, -2, 6, -8, 4, 20, -36, 24, -8, 72, -160, 144, -64, 16, 272, -720, 800, -480, 160, -32, 1056, -3264, 4320, -3200, 1440, -384, 64, 4160, -14784, 22848, -20160, 11200, -4032, 896, -128, 16512, -66560, 118272, -121856, 80640, -35840, 10752, -2048, 256
Offset: 0

Views

Author

Peter Luschny, Jul 12 2019

Keywords

Examples

			[0] [    1]
[1] [    2,      -2]
[2] [    6,      -8,      4]
[3] [   20,     -36,     24,      -8]
[4] [   72,    -160,    144,     -64,     16]
[5] [  272,    -720,    800,    -480,    160,     -32]
[6] [ 1056,   -3264,   4320,   -3200,   1440,    -384,     64]
[7] [ 4160,  -14784,  22848,  -20160,  11200,   -4032,    896,   -128]
[8] [16512,  -66560, 118272, -121856,  80640,  -35840,  10752,  -2048,  256]
[9] [65792, -297216, 599040, -709632, 548352, -290304, 107520, -27648, 4608, -512]

Crossrefs

Cf. A326480, A063376.

Programs

  • Maple
    IE2 := proc(n) (exp(z) + 1)^2/(4*exp(x*z));
series(%, z, 48); 2^n*n!*coeff(%, z, n) end:
for n from 0 to 9 do PolynomialTools:-CoefficientList(IE2(n), x) od;
  • Mathematica
    T[n_, k_] := 2^n n! SeriesCoefficient[(E^z + 1)^2/(4 E^(x z)), {x, 0, k}, {z, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 21 2019 *)

Formula

Generated by 1/A326480(z), where A326480(z) denotes the generating function of A326480 which generates the Euler polynomials of order 2.
Showing 1-6 of 6 results.

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