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.

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

Author

Peter Luschny, Jul 12 2019

Keywords

• sign
• tabl

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.