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.

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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..44.

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.

Example

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]

Maple

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)]);

Mathematica

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 *)

Crossrefs

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.

Sequence in context: A344897 A011202 A085484 * A116585 A230935 A008293

Adjacent sequences: A326477 A326478 A326479 * A326481 A326482 A326483

Keyword

sign,tabl

Author

Peter Luschny, Jul 11 2019

Status

approved