A326481 -id:A326481 - cp's OEIS frontend

A326480 T(n, k) = 2^n * n! * [x^k] [z^n] (4exp(xz))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n. Coefficients of Euler polynomials of order 2.

1, -2, 2, 2, -8, 4, 4, 12, -24, 8, -16, 32, 48, -64, 16, -32, -160, 160, 160, -160, 32, 272, -384, -960, 640, 480, -384, 64, 544, 3808, -2688, -4480, 2240, 1344, -896, 128, -7936, 8704, 30464, -14336, -17920, 7168, 3584, -2048, 256

Offset: 0

Peter Luschny, Jul 11 2019

T(m, n, k) = 2^n * n! * [x^k] [z^n] (2^m*exp(x*z))/(exp(z) + 1)^m are the coefficients of the generalized Euler polynomials (or Euler polynomials of higher order).
The classical case (m=1) is in A004174, this sequence is case m=2. A different normalization for m=1 is given in A058940 and for m=2 in A326485.
Generalized Euler numbers are 2^n*Sum_{k=0..n} T(m, n, k)*(1/2)^k. The classical Euler numbers are in A122045 and for m=2 in A326483.

			Triangle starts:
[0] [     1]
[1] [    -2,       2]
[2] [     2,      -8,     4]
[3] [     4,      12,   -24,      8]
[4] [   -16,      32,    48,    -64,     16]
[5] [   -32,    -160,   160,    160,   -160,     32]
[6] [   272,    -384,  -960,    640,    480,   -384,    64]
[7] [   544,    3808, -2688,  -4480,   2240,   1344,  -896,   128]
[8] [ -7936,    8704, 30464, -14336, -17920,   7168,  3584, -2048,   256]
[9] [-15872, -142848, 78336, 182784, -64512, -64512, 21504,  9216, -4608, 512]

NIST Digital Library of Mathematical Functions, §24.16(i), Higher-Order Analogs (of Bernoulli and Euler Polynomials), Release 1.0.23 of 2019-06-15.

Let E2_{n}(x) = Sum_{k=0..n} T(n,k) x^k. Then E2_{n}(1) = A155585(n+1),
E2_{n}(0) = A326481(n), E2_{n}(-1) = A326482(n), 2^n*E2_{n}(1/2) = A326483(n),
2^n*E2_{n}(-1/2) = A326484(n), [x^n] E2_{n}(x) = A000079(n).
Cf. A004174, A058940, A326485, A122045, A081733.

E2 := proc(n) (4*exp(x*z))/(exp(z) + 1)^2;
series(%, z, 48); 2^n*n!*coeff(%, z, n) end:
ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(E2(n), x), n=0..9)]);

T[n_, k_] := 2^n n! SeriesCoefficient[4 Exp[x z]/(Exp[z]+1)^2, {z, 0, n}, {x, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 15 2019 *)

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

Peter Luschny, Jul 12 2019

sign

For comments see A326480.

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

# The function E2(n) is defined in A326480.
seq(subs(x=-1, E2(n)), n=0..22);

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

Peter Luschny, Jul 12 2019

sign

For comments see A326480.

Bisections (up to signs): A002436 (even), A000816 (odd).

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

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

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

Peter Luschny, Jul 12 2019

sign

For comments see A326480.

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

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

cp's OEIS Frontend

A326480 T(n, k) = 2^n * n! * [x^k] [z^n] (4exp(xz))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n. Coefficients of Euler polynomials of order 2.

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

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Mathematica

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

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Author

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Maple

Formula

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

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Author

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Maple

cp's OEIS Frontend

A326480 T(n, k) = 2^n * n! * [x^k] [z^n] (4*exp(x*z))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n. Coefficients of Euler polynomials of order 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A326480 T(n, k) = 2^n * n! * [x^k] [z^n] (4exp(xz))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n. Coefficients of Euler polynomials of order 2.